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Full text of "The first six books of the Elements of Euclid, in which coloured diagrams and symbols are used instead of letters .."

&, OTll mm /zm llOkSr $fj- ma <a 2KX '■•'■■ JTMs. ^n r.yjOQS ¥\*f El/1 ffi* 3^» ^^ 'ff.tfv ,^/^e*- Library of the University of Toronto Aaa BYRNE'S EUCLID THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID WITH COLOURED DIAGRAMS AND SYMBOLS ? THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID IN WHICH COLOURED DIAGRAMS AND SYMBOLS ARE USED INSTEAD OF LETTERS FOR THE GREATER EASE OF LEARNERS BY OLIVER BYRNE SURVEYOR OF HER MAJESTY'S SETTLEMENTS IN THE FALKLAND ISLANDS AND AUTHOR OF NUMEROUS MATHEMATICAL WORKS LONDON WILLIAM PICKERING 1847 TO THE RIGHT HONOURABLE THE EARL FITZWILLIAM, ETC. ETC. ETC. THIS WORK IS DEDICATED BY HIS LORDSHIP S OBEDIENT AND MUCH OBLIGED SERVANT, OLIVER BYRNE. Digitized by the Internet Archive in 2011 with funding from University of Toronto http://www.archive.org/details/firstsixbooksofeOOeucl SS3 INTRODUCTION. HE arts and fciences have become fo extenfive, that to facilitate their acquirement is of as much importance as to extend their boundaries. Illuftration, if it does not fhorten the time of ftudy, will at leaft make it more agreeable. This Work has a greater aim than mere illuftration ; we do not intro- duce colours for the purpofe of entertainment, or to amuie by certain cottibinations of tint and form, but to affift the mind in its refearches after truth, to increafe the facilities of instruction, and to diffufe permanent knowledge. If we wanted authorities to prove the importance and ufefulnefs of geometry, we might quote every philofopher fince the days of Plato. Among the Greeks, in ancient, as in the fchool of Peftalozzi and others in recent times, geometry was adopted as the beft gymnaftic of the mind. In fact, Euclid's Elements have become, by common confent, the bafis of mathematical fcience all over the civilized globe. But this will not appear extraordinary, if we coniider that this fublime fcience is not only better calculated than any other to call forth the fpirit of inquiry, to elevate the mind, and to ftrengthen the reafoning faculties, but alfo it forms the beft introduction to moft of the ufeful and important vocations of human life. Arithmetic, land-furveying, men- furation, engineering, navigation, mechanics, hydroftatics, pneumatics, optics, phyfical aftronomy, &c. are all depen- dent on the propofitions of geometry. viii INTRODUCTION. Much however depends on the firfr. communication of any fcience to a learner, though the beft and mod eafy methods are feldom adopted. Propositions are placed be- fore a fludent, who though having a fufficient understand- ing, is told juft as much about them on entering at the very threshold of the fcience, as gives him a prepofleffion molt unfavourable to his future fludy of this delightful fubjedl ; or " the formalities and paraphernalia of rigour are fo oftentatioufly put forward, as almoft to hide the reality. Endlefs and perplexing repetitions, which do not confer greater exactitude on the reafoning, render the demonftra- tions involved and obfcure, and conceal from the view of the fludent the confecution of evidence." Thus an aver- fion is created in the mind of the pupil, and a fubjecl fo calculated to improve the reafoning powers, and give the habit of clofe thinking, is degraded by a dry and rigid courfe of inftruction into an uninteresting exercife of the memory. To raife the curiofity, and to awaken the liftlefs and dormant powers of younger minds mould be the aim of every teacher ; but where examples of excellence are wanting, the attempts to attain it are but few, while emi- nence excites attention and produces imitation. The objecT: of this Work is to introduce a method of teaching geome- try, which has been much approved of by many fcientific men in this country, as well as in France and America. The plan here adopted forcibly appeals to the eye, the mofl fenfitive and the moft comprehenfive of our external organs, and its pre-eminence to imprint it fubjecT: on the mind is fupported by the incontrovertible maxim expreffed in the well known words of Horace : — Segnius irritant animus dcmijfa per aurem §>uam quts funt oculis fubjetta fidelibus. A feebler imprefs through the ear is made, Than what is by the faithful eye conveyed. INTRODUCTION. ix All language confifts of reprefentative figns, and thofe figns are the beft which effect their purpofes with the greateft precifion and difpatch. Such for all common pur- pofes are the audible figns called words, which are ftill confidered as audible, whether addreffed immediately to the ear, or through the medium of letters to the eye. Geo- metrical diagrams are not figns, but the materials of geo- metrical fcience, the objecl: of which is to fhow the relative quantities of their parts by a procefs of reafoning called Demonftration. This reafoning has been generally carried on by words, letters, and black or uncoloured diagrams ; but as the ufe of coloured fymbols, figns, and diagrams in the linear arts and fciences, renders the procefs of reafon- ing more precife, and the attainment more expeditious, they have been in this inftance accordingly adopted. Such is the expedition of this enticing mode of commu- nicating knowledge, that the Elements of Euclid can be acquired in lefs than one third the time ufually employed, and the retention by the memory is much more permanent; thefe facts have been afcertained by numerous experiments made by the inventor, and feveral others who have adopted his plans. The particulars of which are few and obvious ; the letters annexed to points, lines, or other parts of a dia- gram are in fadl but arbitrary names, and reprefent them in the demonftration ; inftead of thefe, the parts being differ- ently coloured, are made g to name themfelves, for their forms in correfpond- ing colours represent them in the demonftration. In order to give a bet- ter idea of this fyftem, and A- of the advantages gained by its adoption, let us take a right I. x INTRODUCTION. angled triangle, and exprefs fome of its properties both by colours and the method generally employed. Some of the properties of the right angled triangle ABC, exprejfed by the method generally employed. 1. The angle BAC, together with the angles BCA and ABC are equal to two right angles, or twice the angle ABC. 2. The angle CAB added to the angle ACB will be equal to the angle ABC. 3. The angle ABC is greater than either of the angles BAC or BCA. 4. The angle BCA or the angle CAB is lefs than the angle ABC. 5. If from the angle ABC, there be taken the angle BAC, the remainder will be equal to the angle ACB. 6. The fquare of AC is equal to the fum of the fquares of AB and BC. The fame properties expreffed by colouring the different parts. That is, the red angle added to the yellow angle added to the blue angle, equal twice the yellow angle, equal two right angles. Or in words, the red angle added to the blue angle, equal the yellow angle. ▲ ^ C JK* or The yellow angle is greater than either the red or blue angle. INTRODUCTION. xi ▲ 4- ^B or Ml Zl Either the red or blue angle is lefs than the yellow angle. ▲ 5- pp minus In other terms, the yellow angle made lefs by the blue angle equal the red angle. + That is, the fquare of the yellow line is equal to the fum of the fquares of the blue and red lines. In oral demonstrations we gain with colours this impor- tant advantage, the eye and the ear can be addreffed at the fame moment, fo that for teaching geometry, and other linear arts and fciences, in claries, the fyftem is the beft ever propofed, this is apparent from the examples juft given. Whence it is evident that a reference from the text to the diagram is more rapid and fure, by giving the forms and colours of the parts, or by naming the parts and their colours, than naming the parts and letters on the diagram. Befides the fuperior fimplicity, this fyftem is likewife con- fpicuous for concentration, and wholly excludes the injuri- ous though prevalent practice of allowing the ftudent to commit the demonftration to memory; until reafon, and fact, and proof only make impreflions on the underftanding. Again, when lecturing on the principles or properties of figures, if we mention the colour of the part or parts re- ferred to, as in faying, the red angle, the blue line, or lines, &c. the part or parts thus named will be immediately feen by all in the clafs at the fame inftant ; not fo if we fay the angle ABC, the triangle PFQ^the figure EGKt, and fo on ; xii INTRODUCTION. for the letters mud be traced one by one before the ftudents arrange in their minds the particular magnitude referred to, which often occafions confufion and error, as well as lofs of time. Alfo if the parts which are given as equal, have the fame colours in any diagram, the mind will not wander from the object before it ; that is, fuch an arrangement pre- fents an ocular demonstration of the parts to be proved equal, and the learner retains the data throughout the whole of the reafoning. But whatever may be the advantages of the prefent plan, if it be not fubftituted for, it can always be made a powerful auxiliary to the other methods, for the purpofe of introduction, or of a more fpeedy reminifcence, or of more permanent retention by the memory. The experience of all who have formed fyftems to im- prefs fadts on the understanding, agree in proving that coloured reprefentations, as pictures, cuts, diagrams, &c. are more eafily fixed in the mind than mere fentences un- marked by any peculiarity. Curious as it may appear, poets feem to be aware of this fadl more than mathema- ticians ; many modern poets allude to this vifible fyftem of communicating knowledge, one of them has thus expreffed himfelf : Sounds which addrefs the ear are loft and die In one fhort hour, but thefe which ftrike the eye, Live long upon the mind, the faithful fight Engraves the knowledge with a beam of light. This perhaps may be reckoned the only improvement which plain geometry has received fince the days of Euclid, and if there were any geometers of note before that time, Euclid's fuccefs has quite eclipfed their memory, and even occafioned all good things of that kind to be afilgned to him ; like JEfop among the writers of Fables. It may alfo be worthy of remark, as tangible diagrams afford the only medium through which geometry and other linear INTRODUCTION. xiii arts and fciences can be taught to the blind, this vifible fys- tem is no lefs adapted to the exigencies of the deaf and dumb. Care mult be taken to {how that colour has nothing to do with the lines, angles, or magnitudes, except merely to name them. A mathematical line, which is length with- out breadth, cannot pofiefs colour, yet the jun&ion of two colours on the fame plane gives a good idea of what is meant by a mathematical line ; recoiled: we are fpeaking familiarly, fuch a junction is to be underftood and not the colour, when we fay the black line, the red line or lines, &c. Colours and coloured diagrams may at firft. appear a clumfy method to convey proper notions of the properties and parts of mathematical figures and magnitudes, how- ever they will be found to afford a means more refined and extenfive than any that has been hitherto propofed. We fhall here define a point, a line, and a furface, and demonftrate apropofition in order to fhow the truth of this alfertion. A point is that which has pofition, but not magnitude ; or a point is pofition only, abfiradled from the confideration of length, breadth, and thicknefs. Perhaps the follow- ing defcription is better calculated to explain the nature of a mathematical point to thofe who have not acquired the idea, than the above fpecious definition. Let three colours meet and cover a portion of the paper, where they meet is not blue, nor is it yellow, nor is it red, as it occupies no portion of the plane, for if it did, it would belong to the blue, the red, or the yellow part ; yet it exifts, and has pofition without magnitude, fo that with a little reflection, this June- XIV INTRODUCTION. tioii of three colours on a plane, gives a good idea of a mathematical point. A line is length without breadth. With the afliftance of colours, nearly in the fame manner as before, an idea of a line may be thus given : — Let two colours meet and cover a portion of the paper ; where they meet is not red, nor is it blue ; therefore the junction occu- pies no portion of the plane, and therefore it cannot have breadth, but only length : from which we can readily form an idea of what is meant by a mathematical line. For the purpofe of illu fixation, one colour differing from the colour of the paper, or plane upon which it is drawn, would have been fufficient; hence in future, if we fay the red line, the blue line, or lines, &c. it is the junc- tions with the plane upon which they are drawn are to be underftood. Surface is that which has length and breadth without thicknefs. When we confider a folid body (PQ), we perceive at once that it has three dimenfions, namely : — length, breadth, and thicknefs ; S fuppofe one part of this folid (PS) to be red, and the other part (QR) yellow, and that the colours be diftincr. without commingling, the blue furface (RS) which feparates thefe parts, or which is the fame 2 thing, that which divides the folid without lofs of material, mufr. be without thicknefs, and only poffeffes length and breadth ; R 1 INTRODUCTION. xv this plainly appears from reafoning, limilar to that juft em- ployed in defining, or rather defcribing a point and a line. The propofition which we have felefted to elucidate the manner in which the principles are applied, is the fifth of the firft Book. In an ifofceles triangle ABC, the ° A internal angles at the bafe ABC, ACB are equal, and when the fides AB, AC are produced, the exter- nal angles at the bafe BCE, CBD are alio equal. Produce — — — and make — — — — — Draw «— — — and in we have and ^^ common : and Again in = ^ (B. ,. pr. + .) Z 7 ^ \ , xvi INTRODUCTION. and =: (B. i. pr. 4). But ^ Q. E. D. 5y annexing Letters to the Diagram. Let the equal fides AB and AC be produced through the extremities BC, of the third fide, and in the produced part BD of either, let any point D be aflumed, and from the other let AE be cut off equal to AD (B. 1. pr. 3). Let the points E and D, fo taken in the produced fides, be con- nected by ftraight lines DC and BE with the alternate ex- tremities of the third fide of the triangle. In the triangles DAC and EAB the fides DA and AC are reflectively equal to EA and AB, and the included angle A is common to both triangles. Hence (B. 1 . pr. 4.) the line DC is equal to BE, the angle ADC to the angle AEB, and the angle ACD to the angle ABE ; if from the equal lines AD and AE the equal fides AB and AC be taken, the remainders BD and CE will be equal. Hence in the triangles BDC and CEB, the fides BD and DC are refpectively equal to CE and EB, and the angles D and E included by thofe fides are alfo equal. Hence (B. 1 . pr. 4.) INTRODUCTION. xvii the angles DBC and ECB, which are thofe included by the third fide BC and the productions of the equal fides AB and AC are equal. Alfo the angles DCB and EBC are equal if thofe equals be taken from the angles DCA and EBA before proved equal, the remainders, which are the angles ABC and ACB oppofite to the equal fides, will be equal. Therefore in an ifofceles triangle, &c. Q^E. D. Our object in this place being to introduce the fyftem rather than to teach any particular fet of propofitions, we have therefore feledted the foregoing out of the regular courfe. For fchools and other public places of inftruclion, dyed chalks will anfwer to defcribe diagrams, 6cc. for private ufe coloured pencils will be found very convenient. We are happy to find that the Elements of Mathematics now forms a confiderable part of every found female edu- cation, therefore we call the attention of thofe interefted or engaged in the education of ladies to this very attractive mode of communicating knowledge, and to the fucceeding work for its future developement. We fhall for the prefent conclude by obferving, as the fenfes of fight and hearing can be fo forcibly and inftanta- neously addreffed alike with one thoufand as with one, the million might be taught geometry and other branches of mathematics with great eafe, this would advance the pur- pofe of education more than any thing that might be named, for it would teach the people how to think, and not what to think ; it is in this particular the great error of education originates. XV1U THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. I. A point is that which has no parts. II. A line is length without breadth. III. The extremities of a line are points. IV. A ftraight or right line is that which lies evenly between its extremities. V. A furface is that which has length and breadth only. VI. The extremities of a furface are lines. VII. A plane furface is that which lies evenly between its ex- tremities. VIII. A plane angle is the inclination of two lines to one ano- ther, in a plane, which meet together, but are not in the fame direction. IX. A plane rectilinear angle is the inclina- tion of two ftraight lines to one another, which meet together, but are not in the fame ftraight line. BOOK I. DEFINITIONS. xix A X. When one ftraight line (landing on ano- ther ftraight line makes the adjacent angles equal, each of thefe angles is called a right angle, and each of thefe lines is faid to be perpendicular to the other. XI. An obtufe angle is an angle greater than a right angle. XII. An acute angle is an angle lefs than a right angle. XIII. A term or boundary is the extremity of any thing. XIV. A figure is a furface enclofed on all fides by a line or lines. XV. A circle is a plane figure, bounded by one continued line, called its cir- cumference or periphery ; and hav- ing a certain point within it, from which all ftraight lines drawn to its circumference are equal. XVI. This point (from which the equal lines are drawn) is called the centre of the circle. *•*•• • xx BOOK I. DEFINITIONS. XVII. A diameter of a circle is a flraight line drawn through the centre, terminated both ways in the circumference. XVIII. A femicircle is the figure contained by the diameter, and the part of the circle cut off by the diameter. XIX. * A fegment of a circle is a figure contained by a flraight line, and the part of the cir- \ J cumference which it cuts off. XX. A figure contained by flraight lines only, is called a recti- linear figure. XXI. A triangle is a rectilinear figure included by three fides. XXII. A quadrilateral figure is one which is bounded by four fides. The flraight lines — «— — and !■.■■■ connecting the vertices of the oppofite angles of a quadrilateral figure, are called its diagonals. XXIII. A polygon is a rectilinear figure bounded by more than four fides. BOOK I. DEFINITIONS. xxi XXIV. A triangle whofe three fides are equal, is faid to be equilateral. XXV. A triangle which has only two fides equal is called an ifofceles triangle. XXVI. A fcalene triangle is one which has no two fides equal. XXVII. A right angled triangle is that which has a right angle. XXVIII. An obtufe angled triangle is that which has an obtufe angle. XXIX. An acute angled triangle is that which has three acute angles. XXX. Of four-fided figures, a fquare is that which has all its fides equal, and all its angles right angles. XXXI. A rhombus is that which has all its fides equal, but its angles are not right angles. XXXII. An oblong is that which has all its angles right angles, but has not all its fides equal. u xxii BOOK 1. POSTULATES. XXXIII. A rhomboid is that which has its op- pofite fides equal to one another, but all its fides are not equal, nor its angles right angles. XXXIV. All other quadrilateral figures are called trapeziums. XXXV. Parallel flraight lines are fuch as are in ■'^^^ m ^ mmm ^ m ^ mmi ^ the fame plane, and which being pro- duced continually in both directions, would never meet. POSTULATES. I. Let it be granted that a flraight line may be drawn from any one point to any other point. II. Let it be granted that a finite flraight line may be pro- duced to any length in a flraight line. III. Let it be granted that a circle may be defcribed with any centre at any diflance from that centre. AXIOMS. I. Magnitudes which are equal to the fame are equal to each other. II. If equals be added to equals the fums will be equal. BOOK I. AXIOMS. xxin III. If equals be taken away from equals the remainders will be equal. IV. If equals be added to unequals the fums will be un- equal. V. If equals be taken away from unequals the remainders will be unequal. VI. The doubles of the fame or equal magnitudes are equal. VII. The halves of the fame or equal magnitudes are equal. VIII. Magnitudes which coincide with one another, or exactly fill the fame fpace, are equal. IX. The whole is greater than its part. X. Two flraight lines cannot include a fpace. XI. All right angles are equal. XII. If two ftraight lines ( Z^ZI flraight line (« ) meet a third ■ ) fo as to make the two interior angles ( and i ^ ) on the fame fide lefs than two right angles, thefe two ftraight lines will meet if they be produced on that fide on which the angles are lefs than two right angles. XXIV BOOK I. ELUCIDATIONS. The twelfth axiom may be expreffed in any of the fol- lowing ways : i . Two diverging ftraight lines cannot be both parallel to the fame ftraight line. 2. If a flraight line interfecT: one of the two parallel ftraight lines it muft alfo interfecl the other. 3. Only one flraight line can be drawn through a given point, parallel to a given ftraight line. Geometry has for its principal objects the expofition and explanation of the properties of figure, and figure is defined to be the relation which fubfifts between the boundaries of fpace. Space or magnitude is of three kinds, linear, Juper- ficial, &n&folid. Angles might properly be confideret" as a fourth fpecies of magnitude. Angular magnitude evidently confifts of parts, and muft therefore be admitted to be a fpecies ol quantity The ftudent muft not fuppofe that the magni- tude of an angle is affected by the length of the ftraight lines which include it, and of whofe mutual divergence it is the mea- fure. The vertex of an angle is the point where the fides or the legs of the angle meet, as A. An angle is often defignated by a fingle letter when its legs are the only lines which meet to- gether at its vertex. Thus the red and blue lines form the yellow angle, which in other fyftems would be called the angle A. But when more than two B lines meet in the fame point, it was ne- ceffary by former methods, in order to avoid confufion, to employ three letters to defignate an angle about that point, A BOOK I. ELUCIDATIONS. xxv the letter which marked the vertex of the angle being always placed in the middle. Thus the black and red lines meeting together at C, form the blue angle, and has been ufually denominated the angle FCD or DCF The lines FC and CD are the legs of the angle; the point C is its vertex. In like manner the black angle would be defignated the angle DCB or BCD. The red and blue angles added together, or the angle HCF added to FCD, make the angle HCD ; and fo of other angles. When the legs of an angle are produced or prolonged beyond its vertex, the angles made by them on both fides of the vertex are faid to be vertically oppofite to each other : Thus the red and yellow angles are faid to be vertically oppofite angles. Superpojition is the procefs by which one magnitude may be conceived to be placed upon another, fo as exactly to cover it, or fo that every part of each fhall exactly coin- cide. A line is faid to be produced, when it is extended, pro- longed, or has its length increafed, and the increafe of length which it receives is called its produced part, or its production. The entire length of the line or lines which enclofe a figure, is called its perimeter. The firft fix books of Euclid treat of plain figures only. A line drawn from the centre of a circle to- its circumference, is called a radius. The lines which include a figure are called its Jides. That fide of a right angled triangle, which is oppofite to the right angle, is called the hypotenufe. An oblong is defined in the fecond book, and called a rectangle. All the lines which are conlideied in the firft. fix books of the Elements are fuppofed to be in the fame plane. The Jlraight-edge and compajfes are the only inftruments, xxvi BOOK I. ELUCIDATIONS. the ufe of which is permitted in Euclid, or plain Geometry. To declare this reftriction is the object of the populates. The Axioms of geometry are certain general propofitions, the truth of which is taken to be felf-evident and incapable of being eftabliflied by demonftration. Propofitions are thofe refults which are obtained in geo- metry by a procefs of reafoning. There are two fpecies of propofitions in geometry, problems and theorems. A Problem is a propofition in which fomething is pro- pofed to be done ; as a line to be drawn under fome given conditions, a circle to be defcribed, fome figure to be con- firucted, 5cc. The folution of the problem confifts in fhowing how the thing required may be done by the aid of the rule or ftraight- edge and compafies. The demonjlration confifts in proving that the procefs in- dicated in the folution really attains the required end. A Theorem is a propofition in which the truth of fome principle is aflerted. This principle muft be deduced from the axioms and definitions, or other truths previously and independently eftablifhed. To fhow this is the object of demonstration. A Problem is analogous to a poftulate. A Theorem refembles an axiom. A Pojlulate is a problem, the folution of which is afiumed. An Axiom is a theorem, the truth of which is granted without demonfbration. A Corollary is an inference deduced immediately from a propofition. A Scholium is a note or obfervation on a propofition not containing an inference of fufficient importance to entitle it to the name of a corollary. A Lemma is a propofition merely introduced for the pur- pole of efiablifhing fome more important propofition. XXV11 SYMBOLS AND ABBREVIATIONS. ,*. exprefies the word therefore. V becaufe. — equal. This fign of equality may be read equal to, or is equal to, or are equal to ; but any difcrepancy in regard to the introduction of the auxiliary verbs Is, are, &c. cannot affect the geometri- cal rigour. d\p means the fame as if the words ' not equal' were written. r~ fignifies greater than. 33 ... . lefs than. if ... . not greater than. ~h .... not lefs than. -j- is vezdplus (more), the fign of addition ; when interpofed between two or more magnitudes, fignifies their fum. — is read minus (lefs), fignifies fubtraction ; and when placed between two quantities denotes that the latter is to be taken from the former. X this fign exprefies the product of two or more numbers when placed between them in arithmetic and algebra ; but in geometry it is generally ufed to exprefs a rect- angle, when placed between " two fixaight lines which contain one of its right angles." A reclangle may alfo be reprefented by placing a point between two of its conterminous fides. 2 exprefies an analogy or proportion ; thus, if A, B, C and D, reprefent four magnitudes, and A has to B the fame ratio that C has to D, the propofition is thus briefly written, A : B : : C : D, A : B = C : D, A C ° r B=D. This equality or famenefs of ratio is read, xxviii STMBOLS AND ABBREVIATIONS. as A is to B, fo is C to D ; or A is to B, as C is to D. | fignifies parallel to. _L . . . . perpendicular to. . angle. . right angle. m two right angles. Xi x or I > briefly defignates a point. \ . =, or ^ flgnities greater, equal, or lefs than. The lquare defcribed on a line is concifely written thus, In the fame manner twice the fquare of, is expreffed by 2 2 . def. fignifies definition. pos pojlulate. ax axiom. hyp hypothefis. It may be neceffary here to re- mark, that the hypothefis is the condition affumed or taken for granted. Thus, the hypothefis of the pro- pofition given in the Introduction, is that the triangle is ifofceles, or that its legs are equal. conft confiriiolion. The confiruSlion is the change made in the original figure, by drawing lines, making angles, defcribing circles, &c. in order to adapt it to the argument of the demonftration or the folution of the problem. The conditions under which thefe changes are made, are as indisputable as thofe con- tained in the hypothefis. For inftance, if we make an angle equal to a given angle, thefe two angles are equal by confbruction. Q^ E. D Quod erat demonfirandum. Which was to be demonftrated. CORRIGENDA. xxix Faults to be correSled before reading this Volurne. Page 13, line 9, for def. 7 read def. 10. 45, laft line, for pr. 19 raz^pr. 29. 54, line 4 from the bottom, /or black and red line read blue and red line. 59, line 4, /or add black line fquared read add blue line fquared. 60, line 17, /or red line multiplied by red and yellow line read red line multiplied by red, blue, and yellow line. 76, line 11, for def. 7 read def. 10. 81, line 10, for take black line r*W take blue line. 105, line 11, for yellow black angle add blue angle equal red angle read yellow black angle add blue angle add red angle. 129, laft line, for circle read triangle. 141, line 1, for Draw black line read Draw blue line. 196, line 3, before the yellow magnitude infert M. ©ttclto- BOOK I. • PROPOSITION I. PROBLEM. N a given finite Jlraight line ( ) to defcribe an equila- teral triangle. Defcribe I —J and © (poftulate 3.); draw and — — (port. 1.). then will \ be equilateral. For -^— = (def. 15.); and therefore * \ is the equilateral triangle required. Q^E. D. BOOK I. PROP. II. PROB. ROM a given point ( ■■ ), to draw ajiraight line equal to a given finite Jlraight line ( ). ■- (port, i.), defcribe A(pr. i.), produce — — (port. © 2.), defcribe (poft. 3.), and (poft. 3.) ; produce — — — (poft. 2.), then is the line required. For and (def. 15.), (conft.), ,\ (ax. 3.), but (def. 15.) drawn from the given point ( )> is equal the given line Q. E. D. BOOK I. PROP. III. PROB. ROM the greater ( — ■) of two given Jiraight lines, to cut off a part equal to the lefs ( ) . Draw (pr. 2.) ; defcribe (port. 3 .), then For and (def. 15.), (conft.) ; (ax. 1.). Q. E. D. BOOK I. PROP. IF. THEOR. F two triangles have two Jides of the one reJpecJively equal to two Jides of the other, ( ■ to — ■— and — — to ■ ) tfW //$*• rf«£/<?j ( and ) contained by thofe equal fides alfo equal ; then their bafes or their fdes (■ and — ■ ) are alfo equal : and the remaining and their remain- ing angles oppofte to equal fdes are refpeSlively equal i J^ =z ^^ and ^^ = | f^ ) : and the triangles are equal in every reJpecJ. Let the two triangles be conceived, to be fo placed, that the vertex of the one of the equal angles, or $ — — to coincide — coincide with ■ if ap- will coincide with — ■— — , or two ftraight lines will enclofe a fpace, which is impoflible fliall fall upon that of the other, and r with 9 then will - plied : confequently — — — (ax. 10), therefore > = > and ^L = ^L , and as the triangles / \ and /V coincide, when applied, they are equal in every refpedl:. Q. E. D. BOOK I. PROP. V. THEOR. N any ifofceles triangle A if the equal fides be produced, the external angles at the bafe are equal, and the internal angles at the bafe are alfo equal. Produce ; and (poft. 2.), take j (P r - 3-); draw- Then in common to (conft), ^ (hyp.) /. Jk = |k A = ±,-A=A (pr. 4.) but (ax. 3.) Q.E. D. BOOK I. PROP. VI. THEOR. and N any triangle ( A )if two angles ( and j^L ) are equal \t lie fides ( .... ■"■ ' ) oppojite to them are alfo equal. For if the fides be not equal, let one of them ■■■■ be greater than the other — , and from it cut off ■ = — — ■ — (pr. 3.), draw Then (conft.) in L-^A, (hyp.) and common, .*. the triangles are equal (pr. 4.) a part equal to the whole, which is abfurd ; .'. neither of the fides ■— » or ■ ■ m i is greater than the other, /. hence they are equal Q. E. D. BOOK I. PROP. VII. THEOR. 7 N the fame bafe (> ■), and on the fame fide of it there cannot be two triangles having their conterminous fdes ( and ■- — ■— , — ■— ■« — ■» #«</ ■■»■■■■■») at both extremities of the bafe, equal to each other. When two triangles ftand on the fame bafe, and on the fame fide of it, the vertex of the one (hall either fall outfide of the other triangle, or within it ; or, laflly, on one of its fides. If it be poflible let the two triangles be con- f = 1 firucted fo that draw 0=* J and, (P r - 5-) , then and ▼ => but (pr. 5.) s which is abfurd, therefore the two triangles cannot have their conterminous fides equal at both extremities of the bafe. Q. E. D. BOOK I. PROP. VIII. THEOR. F two triangles have two Jides of the one reflec- tively equal to two fides of the other and — — = ), and alfo their bafes ( rr — ■"■)> equal ; then the angles ("^^B and "^^H') contained by their equal fides are alfo equal. If the equal bafes and be conceived to be placed one upon the other, fo that the triangles fhall lie at the fame fide of them, and that the equal fides «. __» and — _ , _ _____ and _____,__,_. be con- terminous, the vertex of the one muft fall on the vertex of the other ; for to fuppofe them not coincident would contradict the laft propofition. Therefore the fides cident with and „ and A = A ,« being coin- Q. E. D. BOOK I. PROP. IX. PROB. bifeSl a given reSlilinear angle 4 ). Take (P r - 3-) draw , upon which defcribe ^f draw — ^— (pr. i.), Becaufe — — — . = ___ (confl:.) and ^— i — common to the two triangles and (confl:.), A ( P r. 8.) Q. E. D. 10 BOOK I. PROP. X. PROB. O bifefi a given finite Jiraight line ( ««■■■■). and common to the two triangles. Therefore the given line is bifecled. Qj. E. D. BOOK I. PROP. XL PROB. ii ( ; a perpendicular. ROM a given point (^— ™ ')> in a given Jlraight line — ), to draw Take any point (■ cut off ■ ) in the given line, — (P r - 3-)» A conftrucl: £_ \ (pr. i.), draw and it fhall be perpendicular to the given line. For (conft.) (conft.) and common to the two triangles. Therefore Jj ~ (pr. 8.) (def. io.). Q^E. D. 12 500A: /. PROP. XII. PROD. O draw a Jiraight line perpendicular to a given / indefinite Jiraight line («a^_ ) from a given {point /Y\ ) 'without. With the given point x|\ as centre, at one fide of the line, and any diftance — — — capable of extending to the other fide, defcribe Make draw — (pr. 10.) and then For (pr. 8.) fince (conft.) and common to both, = (def. 15.) and (def. io.). Q. E. D. BOOK I. PROP. XIII. THEOR. *3 HEN a Jiraight line ( ) Jlanding upon another Jiraight line ( ) makes angles with it; they are either two right angles or together equal to two right angles. If be J_ to then, and *=0\ (def. 7.), But if draw + + jm = be not JL to , -L ;(pr. 11.) = ( I J (conft.), : mm + V+mk(zx.2.) Q. E. D. H BOOK I. PROP. XIV. THEOR. IF two jlr aight lines ( and "~*"^), meeting a third Jlr aight line ( ), «/ //tf yZras* ^w«/, tfW ^/ oppojite fides of it, make with it adjacent angles ( and A ) equal to two right angles ; thefe fraight lines lie in one continuous fraight line. For, if pomble let j and not be the continuation of then but by the hypothecs ,. 4 = A + (ax. 3.); which is abfurd (ax. 9.). is not the continuation of and the like may be demonftrated of any other flraight line except , .*. ^^— ^— is the continuation of Q. E. D. BOOK I. PROP. XV. THEOR. 15 F two right lines ( and ) interfeSl one another, the vertical an- gles and and <4 are equal. <4 + ► 4 In the fame manner it may be lhown that Q^_E. D. i6 BOO A' /. PROP. XVI. THEOR. F a fide of a trian- \ is produced, the external angle ( V..„\ ) « greater than either of the internal remote angles ( A "- A Make Draw = ------ (pr. io.). and produce it until — : draw - . In \ and ^*^f . ► 4 (conft. pr. 15.), .'. ^m = ^L (pr. 4.), In like manner it can be mown, that if ^^— ■ - be produced, ™ ^ IZ ^^ . and therefore which is = ft is C ^ ft . Q. E. D. BOOK I. PROP. XVII. THEOR. 17 NY tiao angles of a tri- A angle ^___Jk are to- gether lefs than two right angles. Produce A + , then will = £D But CZ Jk (pr. 16.) and in the fame manner it may be mown that any other two angles of the triangle taken together are lefs than two right angles. Q. E. D. D i8 BOOK I. PROP. XVIII. THEOR. A N any triangle if one fide «■■*» be greater than another , the angle op- pofite to the greater fide is greater than the angle oppofite to the lefs. 1. e. * Make (pr. 3.), draw Then will J/i R ~ J| ^ (pr. 5.); but MM d (pr. 16.) ,*. £ ^ C and much more , s ^c >. Q. E. D. BOOK I. PROP. XIX. THEOR. *9 A F in any triangle one angle mm be greater than another J ^ the fide which is oppofite to the greater angle, is greater than the Jide oppofite the lefs. If be not greater than or then mull If then A (p r - 5-) ; which is contrary to the hypothefis. — is not lefs than — — ■ — ; for if it were, (pr. 1 8.) which is contrary to the hypothefis : Q. E. D. 20 BOOK I. PROP. XX. THEOR. 4 I NY two fides and •^^-— of a triangle Z\ taken together are greater than the third fide (■ ')• Produce and draw (P r - 3-); Then becaufe —' (conft.). ^ = 4 (pr - *c4 (ax. 9.) + and .'. + (pr. 19.) Q.E.D BOOK I. PROP. XXL THEOR. 21 •om any point ( / ) A within a triangle Jiraight lines be drawn to the extremities of one fide (_.... ), thefe lines tnujl be toge- ther lefs than the other twofdes, but tnujl contain a greater angle. Produce mm— mm -f- mmmmmm C «-^— ■» (pr. 20.), add ..... to each, -\- __.-.. C ■■— ■ -|- ...... (ax. 4.) In the fame manner it may be mown that .— + C h which was to be proved. Again and alfo 4c4 c4 (pr. 16.), (pr. 16.), QJE.D. 22 BOOK I. PROP. XXII. THEOR. [IVEN three right lines < ■•••■■- the fum of any two greater than the third, to conJlruEl a tri- angle whoje fides fliall be re- fpeSlively equal to the given lines. BOOK I. PROP. XXIII. PROB. 23 T a given point ( ) in a given firaight line (^— ■■— ), to make an angle equal to a given reel i lineal angle (jgKm ) Draw between any two points in the legs of the given angle. fo that Conftruct and A (pr. 22.). Then (pr. 8.). Q. E. D. 24 BOOK I. PROP. XXIV. THEOR. F two triangles have two fides of the one reflec- tively equal to two fides of the other ( to ————— and ------- to ), and if one of the angles ( < jl ^ ) contain- ed by the equal fides be [L m \)> the fide ( — — ^ ) which is oppofite to the greater angle is greater than the fide ( ) which is oppofte to the lefs angle. greater than the other (L. m \), the fide ( Make C3 = / N (pr. 23.), and — ^— = (pr. 3.), draw ■■■■■■■•■- and --•--—. Becaufe — — — =: — — — (ax. 1. hyp. conft.) .'. ^ = ^f (pr - but '^^ Z2 * » .*. ^J Z] £^' /. — — CI (pr. 19.) but ■- = (p r -4-) .-. c Q. E. D. BOOK I. PROP. XXV. THEOR. 25 F two triangles have two Jides ( and — ) of the one refpeSlively equal to two Jides ( and ) of the other, but their bafes unequal, the angle fubtended by the greater bafe (-^— — ) of the one, mujl be greater than the angle fubtended by the lefs bafe ( ) of the other. ▲ A A A = , CZ or Z2 mk > s not equal to ^^ = ^ then ^— — ss (pr. 4.) for if zz ^^ then ■— — « = which is contrary to the hypothefis ; is not lefs than for if A A=A then (pr. 24.), which is alfo contrary to the hypothefis 1= m* Q^E. D. 26 BOOK I. PROP. XXVI. THEOR. Case I. F two triangles have two angles of the one re- fpeflively equal to two angles of tlie other, ( and Case II. AA \), and a fide of the one equal to a fide of the other fimilarly placed with reJpecJ to the equal angles, the remaining fdes and angles are refpeclively equal to one another. CASE I. Let ■ ..!■■ — and ....■■ ■■ which lie between the equal angles be equal, then i^BHI ~ MMMMMItM . For if it be poffible, let one of them greater than the other ; be make In and draw we have A = A (pr. 4.) BOOK I. PROP. XXVI. THEOR. 27 but JA = Mm (hyp. and therefore ^Bl = ■ &. which is abfurd ; hence neither of the fides ■""■""■ and ——•■■• is greater than the other ; and .*. they are equal ; ., and </] = ^j ? (pr< 4.). CASE II. Again, let «^— — • — «— — ^— ? which lie oppofite the equal angles MmL and 4Hk>. If it be poflible, let -, then take — — — ■ =: «- ■ -■" — ■, draw- Then in * ^ and Lm~. we have — = and = , .'. mk. = Mi (pr- 4-) but Mk = mm (hyp-) .*. Amk. = AWL which is abfurd (pr. 16.). Confequently, neither of the fides •— — ■• or ■—•••• is greater than the other, hence they muft be equal. It follows (by pr. 4.) that the triangles are equal in all refpedls. Q. E. D. 28 BOOK I. PROP. XXVII. THEOR. are parallel. F ajlraight line ( ) meet- ing two other Jlraight lines, and ) makes •with them the alternate angles ( and ) equal, thefe two Jlraight lines If be not parallel to they (hall meet when produced. If it be poffible, let thofe lines be not parallel, but meet when produced ; then the external angle ^w is greater than flftk. (pr. 16), but they are alfo equal (hyp.), which is abfurd : in the fame manner it may be fhown that they cannot meet on the other fide ; .*. they are parallel. Q. E. D. BOOK I. PROP. XXVIII. THEOR. 29 F aflraight line ting two other Jlraight lines and ), makes the external equal to the internal and oppojite angle, at the fame fide of the cutting line {namely, ( A A or ), or if it makes the two internal angles at the fame fide ( V ■ and ^^ , or || ^ tfW ^^^) together equal to two right angles, thofe two fraight lines are parallel. Firft, if mL = jik- then A = W A = W • 1 = (pr-'i 5-)» (pr. 27.). A II Secondly, if J| £ -}- | = then ^ + ^F = L— JL. J(pr- i3-)» (ax. 3.) * = ▼ (pr. 27.) BOOK I. PROP. XXIX. THEOR. STRAIGHT line ( ) falling on two parallel Jlraight lines ( and ), makes the alternate angles equal to one another ; and alfo the external equal to the in- ternal and oppofite angle on the fame fide ; and the two internal angles on the fame fide together equal to tivo right angles. For if the alternate angles draw ■, making Therefore and J^ ^ be not equal, Am (p r - 2 3)- (pr. 27.) and there- fore two flraight lines which interfect are parallel to the fame ftraight line, which is impoffible (ax. 12). Hence the alternate angles and are not unequal, that is, they are equal: = J| m. (pr. 15); .*. J| f^ = J^ ^ , the external angle equal to the inter- nal and oppofite on the fame fide : if M ^r be added to both, then + * =£D (P 1 "-^)- That is to fay, the two internal angles at the fame fide of the cutting line are equal to two right angles. Q. E. D. BOOK I. PROP. XXX. THEOR. 3 1 TRAlGHT/mes( mmm " m ) which are parallel to the fame Jlraight line ( ), are parallel to one another. interfed: Then, (=)• = ^^ = Mm (pr. 29.), II (pr. 27.) Q. E. D. 32 BOOK I. PROP. XXXI. PROD. ROM a given point f to draw a Jlr aight line parallel to a given Jlraight line ( ). Draw — ^— • from the point / to any point / in make then — (pr. 23.), - (pr. 27.). Q, E. D. BOOK I. PROP. XXXII. THEOR. 33 F any fide ( ) of a triangle be pro- duced, the external am T 'gle ( ) is equal to the fum of the two internal and oppofite angles ( and ^ Rt, ) , and the three internal angles of every triangle taken together are equal to two right angles. Through the point / draw II (pr- 3 0- Then < ^^^ ( (pr. 29.), (pr. 13.). 4 + Km*. = ^^ (ax. 2.), and therefore (pr. 13.). O. E. D. 34 BOOK I. PROP. XXXIII. THEOR. fRAIGHT lines (- and ) which join the adjacent extremities of two equal and parallel Jlraight * ), are them/elves equal and parallel. Draw the diagonal. (hyp.) and — — common to the two triangles ; = — — , and^J = ^L (pr. 4.); and .". (pr. 27.). Q. E. D. BOOK I. PROP. XXXIV. THEOR. 35 HE oppofite Jides and angles of any parallelogram are equal, and the diagonal ( ) divides it into two equal parts. Since (pr. 29.) and ■— — common to the two triangles. /. \ > (pr. 26.) and m J = m (ax.) : Therefore the oppofite fides and angles of the parallelo- gram are equal : and as the triangles \^ and \^ / are equal in every refpect (pr. 4,), the diagonal divides the parallelogram into two equal parts. Q. E. D. 36 BOOK I. PROP. XXXV. THEOR. ARALLELOGRAMS on the fame bafe, and between the fame paral- lels, are [in area) equal. On account of the parallels, and But, Kpr. 29.) ' (pr- 34-) (pr. 8.) and U minus minus \ ■ Q. E. D. BOOK I. PROP. XXXVI. THEOR. 37 ARALLELO- GRAMS 1 a equal bafes, and between the fame parallels, are equal. Draw and ---..-— , » b y (P r - 34> and hyp.); = and II (pr- 33-) And therefore X is a parallelogram : but !->-■ (P r - 35-) II (ax. 1.). Q. E. D. 38 BOOK I. PROP. XXXVII. THEOR. RIANGLES k and i on the fame bafe (■— «■— ) and bet-ween the fame paral- lels are equal. Draw Produce \ (pr- 3 1 -) L and A and are parallelograms on the fame bafe, and between the fame parallels, and therefore equal, (pr. 35.) ~ twice =r twice i > (P r - 34-) i Q.E D. BOOK I. PROP. XXXVIII. THEOR. 39 RIANGLES II and H ) on : f^wrt/ ^rf/^j and between "•** the fame parallels are equal. Draw ...... and II > (pf - 3 '-'- AM (pr. 36.); i . ,„, 1 but i cs twice ^H (pr. 34.), # i and ^jv = twice ^ (pr. 34.), A A (ax. 7.). Q^E. D. 4o BOOK I. PROP. XXXIX. THEOR. QUAL triangles W \ and ^ on the fame bafe ( ) and on the fame fide of it, are between the fame parallels. If — ■— — » , which joins the vertices of the triangles, be not || — ^— , draw — || (pr.3i.)> meeting ------- . Draw Becaufe II (conft.) (pr- 37-): (hyp.) ; , a part equal to the whole, which is abfurd. -U- — ^— ; and in the fame manner it can be demonftrated, that no other line except is || ; .-. || . O. E. D. BOOK I. PROP. XL. THEOR. QUAL trian- gles 41 ( and L ) on equal bafes, and on the fame Jide, are between the fame parallels. If ..... which joins the vertices of triangles be not 1 1 ■' , draw — — || — — (pr. 31.), meeting Draw Becaufe (conft.) . -- = > , a part equal to the whole, which is abfurd. 1 ' -f|- -^— — : and in the fame manner it can be demonftrated, that no other line except is || : /. || Q. E. D. 42 BOOK L PROP. XLI. THEOR. F a paral- lelogram A V Draw and a triangle are upon the fame bafe ^^^^^— and between the fame parallels ------ and — ^— ^— , the parallelogram is double the triangle. the diagonal ; Then V=J z= twice (P r - 37-) (P r - 34-) ^^ = twice £J . .Q.E.D. BOOK I. PROP. XLII. THEOR. 43 O conflruSl a parallelogram equal to a given 4 triangle ■■■^ L and hav- ing an angle equal to a given rectilinear angle ^ . Make — i^^^— zz ------ (pr. 10.) Draw -. Make J^ = (P n 2 3*) Draw | " jj ~ | (pr. 31.) 4 = twi ce y (pr. 41.) but T = A (pr. 38.) ,V.4. Q. E. D. 44 BOOK I. PROP. XLIII. THEOR. HE complements and ^ ^f of the parallelograms which are about the diagonal of a parallelogram are equal. 1 (pr- 34-) and V = > (pr. 34-) (ax. 3.) Q. E. D. BOOK I. PROP. XLIV. PROB. 45 O a given Jlraight line ( ) to ap- ply a parallelo- gram equal to a given tri- angle ( \ ), and having an angle equal to a given reSlilinear angle Make w. with (pr. 42.) and having one of its fides — — — - conterminous with and in continuation of — ^— — ». Produce -— — till it meets | -•-■•-»• draw prnHnpp it till it mpptc — »—■•» continued ; draw I ■-» meeting produced, and produce -••»•••»• but A=T (pr- 43-J (conft.) (pr.19. and conft.) Q. E. D. BOOK I. PROP. XLV. PROB. O conjlruSl a parallelogram equal to a given reftilinear figure ( ) and having an angle equal to a given reftilinear angle Draw and tl. dividing the rectilinear figure into triangles. Conftruft having = £ (pr. 42.) *~\ #=► and to — — — a ppiy having mW = AW (pr. 44-) man, apply £ =z having HF = AW (P^ 44-) is a parallelogram, (prs. 29, 14, 30.) having ,fl7 = Q. E. D. BOOK I. PROP. XLVI. PROB. 47 PON a given Jlraight line (— ■■ — ) to confiruB a fquare. Draw Draw ■ ing » _L and = (pr. 1 1. and 3.) II drawn • , and meet- W ~W In 1_ M (conft.) S3 a right angle (conft.) M — = a ri g h t angle (pr. 29.), and the remaining fides and angles muft be equal, (pr. 34.) and .*. mk is a fquare. (def. 27.) Q. E. D. 48 BOOK I. PROP. XLVIL THEOR. N a right angled triangle thefquare on the hypotenufe — — — is equal to the fum of the fquares of the fides, (« and ). On and defcribe fquares, (pr. 46.) Draw -■■■ »i alfo draw — — (pr. 31.) - and — ^— . To each add = — -- and Again, becauje BOOK I. PROP. XLVII. THEOR. 49 and := twice twice In the fame manner it may be fhown that # hence ++ Q E. D. H 5° BOOK I. PROP. XLVIIL THEOR. F the fquare of one fide ( — ; — ) of a triangle is equal to the fquares of the other two fides ( n and ), the angle ( )fubtended by that fide is a right angle. Draw — ■ and = (prs.11.3.) ind draw •»•»•■■-•- alfo. Since (conft.) 2 + + but -■ and — 8 + + = — " — ' (P r - 47-). ' = 2 (hyp.) and .*. confequently (pr. 8.), is a right angle. Q. E. D. BOOK II. DEFINITION I. RECTANGLE or a right angled parallelo- gram is faid to be con- tained by any two of its adjacent or conterminous fides. Thus : the right angled parallelogram ■ be contained by the fides ■— ^— and «- or it may be briefly defignated by is faid to If the adjacent fides are equal; i. e. — — — — s ■—■"■"-"■"j then — ^— — • m ■ i which is the expreflion for the redlangle under is a fquare, and is equal to J and ■ or ■ or 52 BOOK II. DEFINITIONS. DEFINITION II. N a parallelogram, the figure compoied of one 01 the paral- lelograms about the diagonal, together with the two comple- ments, is called a Gnomon. Thus and are called Gnomons. BOOK II. PROP. I. PROP,. 53 HE reclangle contained by two Jlraight lines, one of which is divided into any number of parts, i + is equal to the fum of the rectangles contained by the undivided line, and the fever al parts of the divided line. complete the parallelograms, that is to fay, Draw < ...... > (pr. 31.B. i.) ■ =i + l + l I I I + - + Q.E. D. 54 BOOK II. PROP. II. THEOR. Draw I I F a Jlraight line be divided into any two parts ■■ * > 9 the fquare of the whole line is equal to the fum of the rectangles contained by the whole line and each of its parts. + Defcribe parallel to --- (B. i.pr. 46.) (B. i.pr. 31 ) II Q. E. D. BOOK II. PROP. III. THEOR. 55 F a jlraight line be di- vided into any two parts ■ ■ ■■' , the rectangle contained by the whole line and either of its parts, is equal to the fquare of that part, together with the reBangle under the parts. = — 2 + or, Defcribe (pr. 46, B. 1.) Complete (pr. 31, B. 1.) Then + I but and I + In a limilar manner it may be readily mown Q.E.D 56 BOOK II. PROP. IV. THEOR. F a Jlraight line be divided into any two parts > , the fquare of the whole line is equal to the fquare s of the parts, together with twice the rectangle contained by the parts. twice + + Defcribe draw - (pr. 46, B. 1.) ■ (port. 1.), and (pr. 31, B. 1.) 4 + 4,4 (pr. 5, B. 1.), (pr. 29, B. 1.) *,4 BOOK II. PROP. IV. THEOR. 57 E .*. by (prs.6,29, 34. B. 1.) ^^J is a fquare m For the fame reafons r I is a fquare ss ■ " B , B. but e_j = EJ+M+ |+ twice ■■ » ■— ■ . Q. E. D. 58 BOOK II. PROP. V. PROB. F a Jlraight line be divided into two equal parts and alfo — — — -— — into two unequal parts, the rectangle contained by the unequal parts, together with the fquare of the line between the points offeclion, is equal to the fquare of half that line Defcribe (pr. 46, B. 1.), draw ■ and ^ — 11 ) (pr. 3 i,B.i.) I (p. 36, B. 1.) (p. 43, B. 1.) (ax. 2. .. BOOK II. PROP. V. THEOR. 59 but and (cor. pr. 4. B. 2.) 2 (conft.) .*. (ax. 2.) ■ - H + Q. E. D. 6o BOOK II. PROP. VI. THEOR. 1 1 1 1 1 / l^HHMHHUUHHHmHr F a Jlraight line be bifecled ■ and produced to any point — «^»—— , the reSlangle contained by the •whole line fo increafed, and the part produced, together with the fquare of half the line, is equal to the fquare of the line made up of the half and the produced part . — + ^ Defcribe (pr. 46, B. i.)» draw anc (pr. 31, B.i.) (prs. 36, 43, B. 1 ) but z= (cor. 4, B. 2.) A + (conft.ax.2.) t Q. E. D. BOOK II. PROP. VII. THEOR. F a Jiraight line be divided into any two parts mi , the fquares of the whole line and one of the parts are equal to twice the reSlangle contained by the whole line and that part, together with the fquare of the other parts. wmw— — 2 -I- — — 2 ""■■ 61 Defcribe Draw — i , (pr. 46, B. i.)- (pott. 1.), (pr. 31, B. i.)- = I (P r - 43> B. 1.), * to both, (cor. 4, B. 2.) I (cor. 4, B. 2.) I + ■ + — + + + — * = 2 + Q. E. D. 62 BOOK II. PROP. Fill. THEOR. : iy w^ ■ ■ •••■■■I ■iiitiiiiniii ■■■^■■■■■■■m F ajlraight line be divided into any two parts , the fquare of thefum of the whole line and any one of its parts, is equal to four times the reclangle contained by the whole line, and that part together with the fquare of the other part. — + Produce and make Conftrudt. draw (pr. 46, B. 1.); (pr. 7, B. 11.) = 4 — + Q. E. D. BOOK II. PROP. IX. THEOR. 63 F a ftraight line be divided into two equal parts mm — . and alfo into two unequal parts , the fquares of the unequal parts are together double the fquares of half the line, *■ and of the part between the points offeSlion. - + 2 = 2 » + 2 Make 1 Draw — II _L and = or and 4 and draw 9 II —9 (pr. 5, B. 1.) rr half a right angle, (cor. pr. 32, B. 1.) (pr. 5, B. 1.) rs half a right angle, (cor. pr. 32, B. 1.) = a right angle. t hence (prs. 5, 29, B. 1.). aaa^, ■»■■■ '. (prs. 6, 34, B. 1.) + or -J- (pr. 47, B. 1.) + + * Q. E. D. 6 4 BOOK II. PROP. X. THEOR. ! + F a Jlraight line fec7ed and pro- duced to any point — — » 9 thefquaresofthe whole produced line, and of the produced part, are toge- ther double of the fquares of the half line, and of the line made up of the half and pro- duced part. Make and ■5— J_ and = to draw " ■■in., and • •*«■«• i II draw ■ — or — — « ...... 9 (pr. 31, B. 1.); alfo. jk (pr. 5, B. 1.) = half a right angle, (cor. pr. 32, B. 1 .) (pr. 5, B. 1.) = half a right angle (cor. pr. 32, B. 1.) zz a right angle. BOOK II. PROP. X. THEOR. 65 t =^=i half a right angle (prs. 5, 32, 29, 34, B. 1.), and .___ — «...•■■■» ... — ■ — — _..-..., ("prs. 6, 34, B. 1.). Hence by (pr. 47, B. 1.) Q. E. D. K 66 BOOK II. PROP. XL PROB. O divide a given Jlraight line in Juch a manner, that the reft angle contained by the whole line and one of its parts may be equal to the fquare of the other. !■••■• mm** a Defcribe make — I (pr. 46, B. I.), - (pr. 10, B. 1.), draw take (pr. 3, B. 1.), on defcribe (pr. 46, B. 1.), Produce -— Then, (pr. 6, B. 2.) >•«■■■■■ — (port. 2.). + — ' »■■■*, or, l-l >■••» • •■ Q.E. D. BOOK II. PROP. XII. THEOR. 67 N any obtufe angled triangle, thefquare of the fide fubtend- ing the obtufe angle exceeds the fum of the fquares of the fides containing the ob- tufe angle, by twice the rec- tangle contained by either of thefe fides and the produced 'parts of the fame from the obtufe angle to the perpendicular let fall on it from the oppofi'ce acute angle. + * by By pr. 4, B. 2. -„..* = 2 _j 3 _|_ 2 „ . add -^— — 2 to both 2 + 2 = 8 (pr. 47f B.i.) 2 • + + ■ or + (pr. 47, B. 1.). Therefore, »" = 2 • ■ ~ : hence ■ by 2 + - + + Q. E. D. 68 BOOK II. PROP. XIII. THEOR. FIRST SECOND. N any tri- angle, the fqnareofthe Jide fubtend- ing an acute angle, is lefs than the fum of the fquares of the Jides con- taining that angle, by twice the rectangle contained by either of thefe fides, and the part of it intercepted between the foot of the perpendicular let fall on it from the oppofite angle, and the angular point of the acute angle. FIRST. '■ -| 2 by 2 . SECOND. 2 -\ 2 by 2 Firrt, fuppofe the perpendicular to fall within the triangle, then (pr. j, B. 2.) ■»■■■ 2 -J- — ^^ 2 ZZZ 2 • ^"—"m • <^^^ -|- ■■■■«■ ', add to each _ ' then, I 2 1 2 ^^ ~ .*. (pr. 47 » B - O BOOK II. PROP. XIII. THEOR. 69 Next fuppofe the perpendicular to fall without the triangle, then (pr. 7, B. 2.) add to each — — 2 then «■». '-' -j- - -{- — — - — 2 • .... • — — . + „„.= + . /# ( pr . 47 , B. i.), -_«a» ' -|- «^— 2 ~ 2 • •■■■■■■■■ • — — -j- -i ■■ I i e Q. E. D. 7° BOOK II. PROP. XIV. PROB. O draw a right line of which the fquare fliall be equal to a given recJi- linear figure. fuch that, * Make (pr. 45, B. i.), produce take --■■•• until (pr. 10, B. i.), Defcribe and produce — 2 (P°ft. 3-). to meet it : draw Or "■""■■"™ ~~ m »-■»•- • ■ ••••— ^ —I— taamia* (pr. 5, B. 2.), but " zz ii ~ -\- ■•••«■•«- (pr. 47, B. i.); mm*m*mm— — f- ■■•«■• mm ^ «■■■■■■ • ■ mm * ■■■■■■■1 —ft— ««•••#«■ — , and ■■■ • ••■>■ Q. E. D. BOOK III. DEFINITIONS. I. QUAL circles are thofe whofe diameters are equal. II. A right line is said to touch a circle when it meets the circle, and being produced does not cut it. III. Circles are faid to touch one an- other which meet but do not cut one another. IV. Right lines are faid to be equally diflant from the centre of a circle when the perpendiculars drawn to them from the centre are equal. 7 2 DEFINITIONS. V. And the ftraight line on which the greater perpendi- cular falls is faid to be farther from the centre. VI. A fegment of a circle is the figure contained by a ftraight line and the part of the circum- ference it cuts off. VII. An angle in a fegment is the angle con- tained by two ftraight lines drawn from any point in the circumference of the fegment to the extremities of the ftraight line which is the bafe of the fegment. VIII. An angle is faid to ftand on the part of the circumference, or the arch, intercepted between the right lines that contain the angle. IX. A fedtor of a circle is the figure contained by two radii and the arch between them. DEFINITIONS. 73 X. Similar fegments of circles are thofe which contain equal angles. Circles which have the fame centre are called concentric circles. 74 BOOK III. PROP. I. PROB. O find the centre of a given circle o Draw within the circle any ftraight draw — — _L -■■---- • biledt — wmmmmm ■ , and the point of bifecfion is the centre. For, if it be pofhble, let any other point as the point of concourfe of — — — , »■— ■ ' and —«■«■■» be the centre. Becaufe in and \/ —— ss ------ (hyp. and B. i, def. 15.) zr »■■•-—- (conft.) and —■««■■- common, ^B. 1, pr. 8.), and are therefore right angles ; but ym = £2 ( c °»ft-) yy = (ax. 1 1 .) which is abfurd ; tberefore the afTumed point is not the centre of the circle ; and in the fame manner it can be proved that no other point which is not on — — ^— is the centre, therefore the centre is in ' , and therefore the point where < is bifecled is the centre. Q. E. D. BOOK III. PROP. II. THEOR. 75 STRAIGHT line ( ■■ ) joining two points in the circumference of a circle , lies wholly within the circle. Find the centre of o (B. 3 .pr.i.); from the centre draw to any point in meeting the circumference from the centre ; draw and ■ . Then = ^ (B. i.pr. 5.) but or \ (B. i.pr. 16.) - (B. 1. pr. 19.) but .*. every point in lies within the circle. Q. E. D. 76 BOOK III. PROP. III. THEOR. F a jlraight line ( — — ) drawn through the centre of a circle o SifecJs a chord ( •"•) which does not pafs through the centre, it is perpendicular to it; or, if perpendicular to it, it bifeSls it. Draw and to the centre of the circle. In -^ I and | .„^S. common, and « ■ ■ ■ • . • ■ ■ and .'. = KB. i.pr.8.) JL (B. i.def. 7.) Again let ______ _L .--. Then ,. ^d - b*> (B. i.pr. 5.) (hyp-) and ind .*. (B. 1. pr. 26.) bifefts Q. E. D. BOOK III. PROP. IV. THEOR. 77 F in a circle tivojlraight lines cut one another, which do not bath pafs through the centre, they do not hifecJ one another. If one of the lines pafs through the centre, it is evident that it cannot be bifedled by the other, which does not pafs through the centre. But if neither of the lines or pafs through the centre, draw — ■— from the centre to their interfedlion. If . be bifedled, ........ J_ to it (B. 3. pr. 3.) ft = i _^ and if be bifefted, ...... J_ ( B - 3- P r - 3-) and .*. j P^ = ^ ; a part equal to the whole, which is abfurd : .*. — — ■- — and ii do not bifect one another. Q. E. D. 78 BOOK III. PROP. V. THEOR. F two circles interfetl, they have not the © fame centre. Suppofe it poffible that two interfering circles have a common centre ; from fuch fuppofed centre draw to the interfering point, and ^-^^^....... • meeting the circumferences of the circles. (B. i.def. 15.) ...... (B. 1. def. 15.) _.--.-. • a part equal to the whole, which is abfurd : .*. circles fuppofed to interfedt in any point cannot have the fame centre. Q. E. D. BOOK III. PROP. VI. THEOR. 79 F two circles © touch one another internally, they have not the fame centre. For, if it be poffible, let both circles have the fame centre ; from fuch a fuppofed centre draw i cutting both circles, and to the point of contact. Then and «»•»•■■ - (B. i.def. 15.) - (B. i.def. 15.) equal to the whole, which is abfurd ; therefore the afTumed point is not the centre of both cir- cles ; and in the fame manner it can be demonftrated that no other point is. Q. E. D. 8o BOOK III. PROP. VII. THEOR. FIGURE I. FIGURE II. F from any point within a circle which is not the centre, lines are drawn to the circumference ; the greatejl of thofe lines is that (—■•■■■■-) which pajfes through the centre, and the leaf is the remaining part ( — ) of the diameter. Of the others, that ( — — — ) which is nearer to the line pafjing through the centre, is greater than that ( «^ » ) which is more remote. Fig. 2. The two lines (' and ) which make equal angles with that paffing through the centre, on oppoftefdes of it, are equal to each other; and there cannot be drawn a third line equal to them, from the fame point to the circumference. FIGURE I. To the centre of the circle draw ------ and «-■■■—• then ------ — -.- (B. i. def. 15.) vmmwmmmam = — — -j- ■-■ C — — — (B.I. pr. 20.) in like manner ■■« .1 ±1 may be fhewn to be greater than M 1 ■ ; or any other line drawn from the fame point to the circumference. Again, by (B. 1. pr. 20.) take — — from both ; .*. — — — C (ax.), and in like manner it may be fhewn that is lefs BOOK III. PROP. VII. THEOR. 81 than any other line drawn from the fame point to the cir- cumference. Again, in **/ and common, m £2 ? anc ^ (B. i. pr. 24.) and may in like manner be proved greater than any other line drawn from the fame point to the circumference more remote from — ^■m—— «. FIGURE II. If ^^ rz then .... — ■ , if not take — — = — — — draw , then s^ I A , -y in ^^ I and , ■ common, = and (B. i.pr. 4.) a part equal to the whole, which is abfurd : — — =1 *■■■■»..*.; and no other line is equal to — drawn from the fame point to the circumfer- ence ; for if it were nearer to the one paffing through the centre it would be greater, and if it were more remote it would be lefs. Q. E. D. M 82 BOOK III. PROP. Fill. THEOR. The original text of this propofition is here divided into three parts. F from a point without a circle, Jlraight f: lines are drawn to the cir- cumference ; of thofe falling upon the concave circum- ference the greatejl is that (— ^.-«.) which pajfes through the centre, and the line ( ' " ) ^hich is nearer the greatejl is greater than that ( ) which is more remote. Draw -■-■•••••• and •■■■••■■■■ to the centre. Then, ■— which palTes through the centre, is greateit; for fince — — ™ = --- . if — ^— ^— be added to both, -■■» :=z •■ ^"™" -p **" ? but [Z (B. i. pr. 20.) .*. ^— « - is greater than any other line drawn from the fame point to the concave circumference. Again in and BOOK III. PROP. VIII. THEOR and i common, but ^ CZ (B. i. pr. 24.); and in like manner may be fhewn C than any other line more remote from II. Of thofe lines falling on the convex circumference the leaf is that (———■-) which being produced would pafs through the centre, and the line which is nearer to the leaf is lefs than that which is more remote. For, lince — — -\~ and ciiitiifl '. And fo of others III. Alfo the lines making equal angles with that which paff'es through the centre are equal, whether falling on the concave or convex circumference ; and no third line can be drawn equal to them from the fame point to the circumference. For if ■■■ make r~ -»■•■■ 9 but making rr L ; = ■■■»■■ ? and draw ■■■■■■ - , 84 BOOK III. PROP. Fill. THEOR. Then in > and / we have and L A common, and alio ^ = , - = (B. i. pr. 4.); but which is abfurd. .....<>... is not :z: --_ * •>■> ■■■■•■■ nor to any part of -...-___ 9 /. ■■■ ■ is not CZ —-----. Neither is ■•• ■• C ■•"•■— ~, they are .*. = to each other. And any other line drawn from the fame point to the circumference mull lie at the fame fide with one of thefe lines, and be more or lefs remote than it from the line pair- ing through the centre, and cannot therefore be equal to it. Q. E. D. BOOK III. PROP. IX. THEOR. 85 F a point b" taken . within a from which ctr„ie o wore than two equal ftraight lines can be drawn to the circumference, that point mujl be the centre of the circle. For, if it be fuppofed that the point |^ in which more than two equal ftraight lines meet is not the centre, lbme other point — '- mult be; join thefe two points by and produce it both ways to the circumference. Then fince more than two equal ftraight lines are drawn from a point which is not the centre, to the circumference, two of them at leaft muft lie at the fame fide of the diameter 'j and fince from a point A, which is not the centre, ftraight lines are drawn to the circumference ; the greateft is ^— ■•■ », which paffes through the centre : and — «~— which is nearer to »«~« ? r~ — — — which is more remote (B. 3. pr. 8.) ; but = (hyp-) which is abfurd. The fame may be demonftrated of any other point, dif- ferent from / \ 9 which muft be the centre of the circle, Q. E. D. 86 BOOK III. PROP. X. THEOR. NE circle I ) cannot inter fe£i another rv J in more points than two. For, if it be poflible, let it interfedt in three points ; from the centre of I J draw O to the points of interferon ; (B. i. def. 15.), but as the circles interfec~t, they have not the fame centre (B. 3. pr. 5.) : .*. the affumed point is not the centre of ^ J , and O and are drawn from a point not the centre, they are not equal (B. 3. prs. 7, 8) ; but it was mewn before that they were equal, which is abfurd ; the circles therefore do not interfedt. in three points. Q. E. D. BOOK III. PROP. XL THEOR. 87 O F two circles and I 1 touch one another internally, the right line joining their centres, being produced, jliall pafs through a point of contact. For, if it be poffible, let join their centres, and produce it both ways ; from a point of contact draw 11 to the centre of f J , and from the fame point of contadl draw •■■•■■•«• to the centre of I I. k Becaufe in +- (B. 1. pr. 20.), I "••!•••••, and O as they are radii of 88 BOOK III. PROP. XL THEOR. but — — -|" — — — C — ; t ak e away — ^— ^ which is common, and -^— ^ d ; but — ^— = -- — • becaufe they are radii of O and .*. CZ a part greater than the whole, which is abfurd. The centres are not therefore fo placed, that a line joining them can pafs through any point but a point of contact. Q. E. D. BOOK III. PROP. XII. THEOR. 89 F two circles o titer externally, the Jlraight line ——■■i»— - - joining their centres, pajfes through the point of contact. touch one ano If it be poffible, let join the centres, and not pafs through a point of contact; then from a point of contact draw and to the centres. Becaufe and « and - + (B. 1. pr. 20.), = (B. 1. def. 15.), = (B. i.def.15.), + , a part greater than the whole, which is abfurd. The centres are not therefore fo placed, thai «"he line joining them can pafs through any point but the point of contact. Q. E. D. N 9 o BOOK III. PROP. XIII. THEOR. FIGURE I. FIGURE II. NE circle can- not touch ano- ther, either externally or internally, in more points than one. FIGURE III. Fig. i . For, if it be poffible, let and f j touch one another internally in two points ; draw ... . i. joining their cen- tres, and produce it until it pafs through one of the points of contadl (B. 3. pr. 11.); draw — — ^— and ~ ^— ^— , But = (B. 1. def. 15.), .*. if be added to both, + but and .*. + + which is abfurd. (B. 1. def. 15.), = — — ; but — (B. 1. pr. 20.), BOOK III. PROP. XIII. THEOR. ot Fig. 2. But if the points of contact be the extremities of the right line joining the centres, this ftraight line mull be bifedled in two different points for the two centres ; be- caufe it is the diameter of both circles, which is abfurd. Fig. 3. Next, if it be pomble, let O and O touch externally in two points; draw ——....-. joining the centres of the circles, and pamng through one of the points of contact, and draw — — — ■ and ^^—^— . — = (B. 1. def. 15.); and ------- — — — — (B. 1. def. 15.): + — — — = — — — ; but + — — • [Z — — (B. 1. pr. 20.), which is abfurd. There is therefore no cafe in which two circles can touch one another in two points. Q E. D. 9 2 BOOK III. PROP. XIV. THEOR. QUALfraight lines (^ ") infcribed in a circle are e- qually diji ant from the centre ; and alfo t Jiraight lines equally dijlant from the centre are equal. From the centre of o draw to ■■■» and ---•-> , join ■-■^— and — — Then and hnce = half (B. 3. pr. 3.) = 1 — ( B - 3- P r -3-) = ..... (hyp.) and (B. i.def. 15.) and but iince is a right angle + ' ' (B.i.pr.47.) ,... 2 -|- M , 2 for the - 2 + fame reafon, + BOOK III. PROP. XIV. THEOR. 93 t ....«<.« • » Alfo, if the lines ....... and ........ be equally diftant from the centre ; that is to fay, if the per- pendiculars -■■ •«-•■- and .......... be given equal, then For, as in the preceding cafe, 1 + 2 = 2 + but ■■amuin " ^Z ■■•■•■■■« " = g , and the doubles of thefe i. and •«_,.... are alfo equal. Q. E. D. 94 BOOK III. PROP. XV. THEOR. FIGURE I. but HE diameter is the greatejl jlraight line in a circle : and, of all others, that which is nearejl to the centre is greater than the more remote. FIGURE I. The diameter — — — is C any line For draw > — — — and —— < and ■ ■ = • — I— i (B. i . pr. 20.) Again, the line which is nearer the centre is greater than the one more remote. Firft, let the given lines be — and , winch are at the fame fide of the centre and do not interfedl ; draw s \ BOOK III. PROP. XV. THEOR. 95 In and \ ► and •■ but \/ and (B. I. pr. 24.) FIGURE II. Let the given lines be — — and — — > which either are at different fides of the centre, or interfec~t ; from the centre draw ......—— and ------ _L and 9 make ........ zz -••--, and draw — — — J_ >— •-— . FIGURE II. Since and the centre, but ■ are equally diftant from (B. 3. pr. 14.); [Pt. i.B. 3. pr. 15.), Q. E. D. 9 6 BOOK III. PROP. XVI. THEOR. HEJlraight line ■ drawn from the extremity of the diame- ter i of a circle perpendicular to it falls *'•... ., without the circle. Jl.*''*" * And if any Jlraight line -■■■■■■■ be drawn from a point i within that perpendi- cular to the point of contact, it cuts the circle. PART I If it be poffible, let which meets the circle again, be J_ ', and draw Then, becauie ^ = ^ (B.i.pr. 5 -), and .*. each of these angles is acute. (B. i. pr. 17.) but = _j (hyp.), which is abfurd, therefore _____ drawn _L — — — - does not meet the circle again. BOOK III. PROP. XVI. THEOR. 07 PART II. Let be J_ — — ■^ and let ------ be drawn from a point *•" between and the circle, which, if it be poflible, does not cut the circle. Becaufe | i = | _j > ^ is an acute angle ; fuppofe ............... J_ ........ 9 drawn from the centre of the circle, it mull: fall at the fide of ^ the acute angle. .*. m^> which is fuppofed to be a right angle, is C Ik , but •«■•»•••«••. ~ — — ■— ■ . and .'. --■•■•>. C -•••••■••■■■■, a part greater than the whole, which is abfurd. Therefore the point does not fall outfide the circle, and therefore the ftraight line ........... cuts the circle. Q.E.D. 98 BOOK III. PROP. XVII. THEOR. O draw a tangent to a given circle f rom a o given point, either in or outjide of its circumference. If the given point be in the cir- cumference, as at „.„| , it is plain that the ftraight line ' mmm "™ J_ — — — the radius, will be the required tan- gent (B. 3. pr. 16.) But if the given point outfide of the circumference, draw — be from it to the centre, cutting draw concentric with then o ( J; and - , defcribe radius zz •■— , will be the tangent required. BOOK III. PROP. XVII. THEOR. zx - A 99 For in __ zz •■-•-■ ■— , jttk common, and (•■•■■■■■•■ ~ ----«■--. (B. i. pr. 4.) = = a right angle, .*. — — — • is a tangent to o ioo BOOK III. PROP. XVIII. THEOR. F a right line •-..... fa a tangent to a circle, the fir aight line — ■ — drawn from the centre to the point of contatt, is perpendicular to it. For, if it be pomble, let ™ ^™" •••■ be _]_ -■••• then becaufe 4 = ^ is acute (B. i . pr. 17.) C (B. 1. pr. 19.); but and .*. — — ■ — - £2 — i the whole, which is abfurd. ►•►••• , a part greater than .". — — is not _L ----- ; and in the fame man- ner it can be demonitrated, that no other line except — ■ — — is perpendicular to ■■■■■ Q. E. D. BOOK III PROP. XIX. THEOR. 101 F a Jlraight line mmKmmmm ^ m be a tangent to a circle, the Jlraight line » , drawn perpendicular to it from point of the contact, pajfes through the centre of the circle. For, if it be poifible, let the centre be without and draw ■ ••■ from the fuppofed centre to the point of contact. Becaufe (B. 3. pr. 18.) = 1 1 , a right angle ; but ^^ = I 1 (hyp.), and ,\ = a part equal to the whole, which is abfurd. Therefore the arTumed point is not the centre ; and in the fame manner it can be demonftrated, that no other point without m ^ mm ^ m is the centre. Q. E. D. 102 BOOK III. PROP. XX. THEOR. FIGURE I HE angle at the centre of a circle, is double the angle at the circumference, when they have the fame part of the circumference for their bafe. FIGURE I. Let the centre of the circle be on ■ ..... a fide of Becaufe k = \ But (B. i. pr. 5.). or + := twice (B. 1. pr. 32). FIGURE 11. FIGURE II. Let the centre be within circumference ; draw ^— 4 j the angle at the from the angular point through the centre of the circle ; ^ = A then ^ = W 9 a °d = , becaufe of the equality of the fides (B. 1. pr. 5). BOOK III. PROP. XX. THEOR. 103 Hence _i_ 4 + + = twke 4 But ^f = 4 + V 9 and twice FIGURE III. Let the centre be without ▼ and __— . the diameter. FIGURE III. draw Becaufe = twice := twice ▲ ZZ twice (cafe 1.) ; and Q. E. D. io4 BOOK III. PROP. XXI. THEOR. FIGURE I. HE angles ( 4& 9 4^ ) in the fame fegment of a circle are equal. FIGURE I. Let the fegment be greater than a femicircle, and draw — ^— ^^— and — — — — to the centre. twice 4Pt or twice ;n (B. 3. pr. 20.) ; 4=4 4 FIGURE II. FIGURE II. Let the fegment be a femicircle, 01 lefs than a femicircle, draw — ■— — ■ the diameter, alfo draw < = 4 > = * (cafe 1.) Q. E. D. BOOK III. PROP. XXII. THEOR. 105 f FIE oppofite angles Afc and ^ j «l «"/,/ o/~ tf«y quadrilateral figure in- ferred in a circle, are together equal to two right angles. Draw and the diagonals ; and becaufe angles in the fame fegment are equal ^r — JP^ and ^r = ^f ; add ^^ to both. two right angles (B. 1. pr. 32.). In like manner it may be fhown that, Q. E. D. io6 BOOK III. PROP. XXIII. THEOR. PON the fame Jlraight line, and upon the fame fide of it, two fimilar fegments of cir- cles cannot he conflrutled which do not coincide. For if it be poffible, let two fimilar fegments Q and be constructed ; draw any right line draw . cutting both the fegments, and — . Becaufe the fegments are fimilar, (B. 3. def. 10.), but (Z ^^ (B. 1. pr. 16.) which is abfurd : therefore no point in either of the fegments falls without the other, and therefore the fegments coincide. O. E. D. BOOK III PROP. XXIV. THEOR. 107 IMILAR fegments and 9 of cir- cles upon equal Jlraight lines ( •— ^— ■ and » ) are each equal to the other. For, if 'j^^ 1^^ be fo applied to that — — — — may fall on , the extremities of — — — may be on the extremities — ^-^— and at the fame fide as becaufe muft wholly coincide with and the fimilar fegments being then upon the fame ftraight line and at the fame fide of it, muft alfo coincide (B. 3. pr. 23.), and are therefore equal. Q. E. D. io8 BOOK III. PROP. XXV. PROB. SEGMENT of a circle being given, to defcribe the circle of which it is the fegment. From any point in the fegment draw mmmmmmmm and — — — bifedl them, and from the points of bifecfion draw -L — ■ — ■ — — and — ■— — — i J- ™^™^^ where they meet is the centre of the circle. Becaufe __ — _ terminated in the circle is bifecled perpendicularly by - , it paffes through the centre (B. 3. pr. I.), likewile — _ paffes through the centre, therefore the centre is in the interferon of thefe perpendiculars. CLE. D. BOOK III. PROP. XXVI. THEOR. 109 N equal circles the arcs O w o on 'which Jland equal angles, •whether at the centre or circum- ference, are equal. Firft, let draw at the centre, and — Then fince OO .«• an d ^VC...........*,';^ have and But k=k (B. 1. pr. 4.). (B. 3-pr. 20.); • O and o are fimilar (B. 3. def. 10.) ; they are alio equal (B. 3. pr. 24.) no BOOK III. PROP. XXVI. THEOR. If therefore the equal fegments be taken from the equal circles, the remaining fegments will be equal ; lence (ax. 3.); and .*. But if the given equal angles be at the circumference, it is evident that the angles at the centre, being double of thofe at the circumference, are alfo equal, and there- fore the arcs on which they ftand are equal. Q. E. D. BOOK III. PROP. XXVII. THEOR. 1 1 1 N equal circles, oo the angles ^v and k which Jland upon equal arches are equal, whether they be at the centres or at the circumferences. For if it be poflible, let one of them ▲ be greater than the other and make k=k ▲ .*. N*_^ = Sw* ( B - 3- P r - 26.) but V^^ = ♦♦.....,.♦ (hyp.) .". ^ , -* = V Lj d/ a part equal to the whole, which is abfurd ; .*. neither angle is greater than the other, and .*. they are equal. Q.E.D *••■■■••• ii2 BOOK III. PROP. XXVIII. TIIEOR. N equal circles equa o-o iitil chords arches. cut off equal From the centres of the equal circles, draw -^^— , — — — and ■ ■■■■■■■■■■ ■ , «■■■■ and becaufe = alib (hyp.) (B. 3. pr. 26.) and .0=0 (ax. 3.) Q. E. D. BOOK III. PROP. XXIX. THEOR. 113 N equal circles O w O the chords — ^— and tend equal arcs are equal. which fub- If the equal arcs be femicircles the propofition is evident. But if not, let and ■5 . anu , be drawn to the centres ; becaufe and but and (hyp-) (B-3.pr.27.); — .......... and -« •-• (B. 1. pr. 4.); but thefe are the chords fubtending the equal arcs. Q. E. D. ii4 BOOK III. PROP. XXX. PROB. O bifecl a given arc C) Draw make draw Draw ■■■-« , and it bifedls the arc. and — — — — . and (conft.), is common, (conft.) (B. i. pr. 4.) = ,*■-%■ (B. 3. pr. 28.), and therefore the given arc is bifedred. Q. E. D. BOOK III. PROP. XXXI. THEOR. 115 N a circle the angle in afemicircle is a right angle, the angle in a fegment greater than a femicircle is acute, and the angle in a feg- ment lefs than afemicircle is obtufe. FIGURE I. FIGURE I. The angle ^ in a femicircle is a right angle. V Draw and JB = and Mk = ^ (B. 1. pr. 5.) + A= V the half of two right angles = a right angle. (B. 1. pr. 32.) FIGURE II. The angle ^^ in a fegment greater than a femi- circle is acute. ▲ Draw the diameter, and = a right angle ▲ is acute. FIGURE II. n6 BOOK III. PROP. XXXI. THEOR. FIGURE III. FIGURE III. The angle v ^k in a fegment lefs than femi- circle is obtufe. Take in the oppofite circumference any point, to which draw — -«— — — and ■■ . * Becaufe -f- (B. 3. pr. 22.) = m but a (part 2.), is obtufe. Q. E. D. BOOK III. PROP. XXXII. THEOR. i F a right line ■—■— — be a tangent to a circle, and from the point of con- tact a right line — — — - be drawn cutting the circle, the angle I made by this line with the tangent is equal to the angle in the alter- ate fegment of the circle. If the chord fhould pafs through the centre, it is evi- dent the angles are equal, for each of them is a right angle. (B. 3. prs. 16, 31.) But if not, draw from the point of contact, it muft pafs through the centre of the circle, (B. 3. pr. 19.) w + f = zLJ = f (b. i.pr.32.) = (ax.). Again O =£Dk= +4 (B. 3. pr. 22.) a-* = ^m , (ax.), which is the angle in the alternate fegment. Q. E. D. 1 1 8 BOOK III. PROP. XXXIII. PROB. N agivenjlraight line — — to dejcribe a fegment of a circle that Jhall contain an angle equal to a given angle ^a, If the given angle be a right angle, bifedl the given line, and defcribe a femicircle on it, this will evidently contain a right angle. (B. 3. pr. 31.) If the given angle be acute or ob- tufe, make with the given line, at its extremity, , draw and make with = ^ , defcribe I I — or as radius, for they are equal. is a tangent to o (B. 3. pr. 16.) divides the circle into two fegments capable of containing angles equal to l W and which were made refpedlively equal ■o£7 and (B. 3 .pr. 32.) Q. E. D. BOOK III. PROP. XXXIV. PROB. 119 O cut off from a given cir- cle I 1 a fegment o which Jljall contain an angle equal to a given angle Draw — (B. 3. pr. 17.), a tangent to the circle at any point ; at the point of contact make the given angle ; contains an angle := the given angle. V Becaufe ■ is a tangent, and — ^—m m cuts it, the ingle angle in (B. 3. pr. 32.), but (conft.) Q. E. D. 120 BOOK III. PROP. XXXV. THEOR. FIGURE I. FIGURE II. F two chords circle I ... .--^_ I tn a cir interject each other, the recJangle contained by the fegments of the one is equal to the re El angle contained by the fegments of the other. FIGURE I. If the given right lines pafs through the centre, they are bifedled in the point of interfedtion, hence the rectangles under their fegments are the fquares of their halves, and are therefore equal. FIGURE II. Let —■»——■— pafs through the 'centre, and __..... not; draw and . Then X (B. 2. pr. 6.), or X x = (B. 2. pr. 5.). X FIGURE III. FIGURE III. Let neither of the given lines pafs through the centre, draw through their interfection a diameter and X = X ...... (Part. 2.), alfo - - X = X (Part. 2.) ; X X Q. E. D. BOOK III. PROP. XXXVI. THEOR. 121 F from a point without a FIGURE I. circle twojiraight lines be drawn to it, one of which — mm is a tangent to the circle, and the other ^— —— . cuts it ; the rectangle under the whole cutting line — «■•" and the external fegment — is equal to the fquare of the tangent — — — . FIGURE I. Let —.-"•• pafs through the centre; draw from the centre to the point of contact ; minus 2 (B. 1. pr. 47), -2 or minus •~~ ^ HH (Liitf BMMW ^Q (B. 2. pr. 6). FIGURE II. If •"••■ do not pafs through the centre, draw FIGURE II. and — — -■ , Then minus " (B. 2. pr. 6), that is, - X minus % ,* (B. 3 .pr. 18). Q. E. D. 122 BOOK III. PROP. XXXVII. THEOR. F from a point out fide of a circle twojlraight lines be drawn, the one ^^— cutting the circle, the other — — — meeting it, and if the recJangle contained by the whole cutting line ■ ■' • and its ex- ternal fegment »-• — •• be equal to thejquare of the line meeting the circle, the latter < is a tangent to the circle. Draw from the given point ___ j a tangent to the circle, and draw from the centre , .....••••, and — ■■--— - ? * = X (fi.3-pr.36-) but ___ 2 = — X — — — (hyp.), and .*. Then in and — — and J and .*■«»— and is common, but and .'. ^ = (B. i.pr. 8.); ZS L_j a right angle (B. 3. pr. 18.), a right angle, is a tangent to the circle (B. 3. pr. 16.). Q. E. D. BOOK IV. DEFINITIONS. RECTILINEAR figure is faid to be infcribedin another, when all the angular points of the infcribed figure are on the fides of the figure in which it is faid to be infcribed. II. A figure is faid to be defcribed about another figure, when all the fides of the circumfcribed figure pafs through the angular points of the other figure. III. A rectilinear figure is faid to be infcribed in a circle, when the vertex of each angle of the figure is in the circumference of the circle. IV. A rectilinear figure is faid to be cir- cumfcribed about a circle, when each of its fides is a tangent to the circle. 124 BOOK IF. DEFINITIONS. A circle is faid to be infcribed in a rectilinear figure, when each fide of the figure is a tangent to the circle. VI. A circle is faid to be circum- fcribed about a rectilinear figure, when the circumference panes through the vertex of each angle of the figure. ¥ is circumfcribed. VII. A straight line is faid to be infcribed in a circle, when its extremities are in the circumference. The Fourth Book of the Elements is devoted to the folution of problems, chiefly relating to the infcription and circumfcrip- tion of regular polygons and circles. A regular polygon is one whofe angles and fides are equal. BOOK IF. PROP. I. PROP,. 125 N a given circle O to place ajlraight line, equal to agivenfiraight line ( ), not greater than the diameter of the circle. Draw -..i-..*— 5 the diameter of ; and if - — z= , then the problem is folved. But if — — ■— « — be not equal to 9 — iz ( h yp-); make -«»«.....- — — — (B. 1. pr. 3.) with ------ as radius, defcribe f 1, cutting , and draw 7 which is the line required. For — ZZ ■••■•»■■•■ — —~ mmmm ^ (B. 1. def. 15. conft.) Q. E. D. 126 BOOK IF. PROP. II. PROB. N a given circle O to tn- fcribe a triangle equiangular to a given triangle. To any point of the given circle draw - , a tangent (B. 3. pr. 17.); and at the point of contact make A m = ^^ (B. 1. pr. 23.) and in like manner draw — , and Becaufe and J^ = ^ (conft.) j£ = ^J (B. 3. pr. 32.) .\ ^^ = ^P ; alfo \/ 5S for the fame reafon. /. ▼ = ^ (B. i.pr. 32.), and therefore the triangle infcribed in the circle is equi- angular to the given one. Q. E. D. BOOK IV. PROP. III. PROB. 12,7 BOUT a given circle O to circumfcribe a triangle equi- angular to a given triangle. Produce any fide , of the given triangle both ways ; from the centre of the given circle draw any radius. Make = A (B. 1. pr. 23.) and At the extremities of the three radii, draw and — — .— ? tangents to the given circle. (B. 3. pr. 17.) The four angles of Z. 9 taken together, are equal to four right angles. (B. 1. pr. 32.) 128 BOOK IV. PROP. III. PROB. but | and ^^^ are right angles (conft.) , two right angles but 4 = L_-l_Ji (^- '■ P r - I 3-) and = (conft.) % and .*. In the fame manner it can be demonstrated that &=a-. 4 = 4 (B. i. pr. 32.) and therefore the triangle circumfcribed about the given circle is equiangular to the given triangle. Q, E. D. BOOK IV. PROP. IV. PROB. 1 2Q N a given triangle A to in- fer i be a circle. Bifedl J and ^V. (B. i.pr. 9.) by and •— ■ ^— from the point where thefe lines meet draw --■-■■■ ? and •••■• refpectively per- pendicular to — — — — , and y 1 In M A'"' > common, .*. ~ ■■ and - *•— (B. 1. pr. 4 and 26.) In like manner, it may be mown alfo that ..—.—..- = — - , ■*#•••»•■•• hence with any one of thefe lines as radius, defcribe and it will pafs through the extremities of the o other two ; and the fides of the given triangle, being per- pendicular to the three radii at their extremities, touch the circle (B. 3. pr. 16.), which is therefore inferibed in the given circle. Q. E. I). 13° BOOK IV. PROP. V. PROB. O defcribe a circle about a given triangle. and ........ (B. i . pr. 10.) From the points of bifedtion draw _L «— ^— and — -— — — — and — refpec- tively (B. i. pr. 11.), and from their point of concourfe draw — — — , •■«■•■■-— and and defcribe a circle with any one of them, and it will be the circle required. In (confl.), common, 4 (conft.), (B. i.pr. 4 .)- ■■■■■a ••■■>» In like manner it may be fhown that a # ..■....■.. ^iz ^^^^^■^■■^ — — "^^^~ \ and therefore a circle defcribed from the concourfe of thefe three lines with any one of them as a radius will circumfcribe the given triangle. Q. E. D. BOOK IV. PROP. VI. PROP,. 131 O N a given circle ( J to infer ibe afquare. Draw the two diameters of the circle _L to each other, and draw . — — , — — and — s> is a fquare. For, fince and fl^ are, each of them, in a femicircle, they are right angles (B. 3. pr. 31), (B. i.pr. 28) and in like manner — — — II And becaufe fl — ^ (conft.), and «•••»»••»•« zzz >■■■■■■■■■» g — »■■•■•■■•■■• (B. 1. def icV .*. = (B. i.pr. 4); and fince the adjacent fides and angles of the parallelo- gram S X are equal, they are all equal (B. 1 . pr. 34) ; o and .*. S ^ ? inferibed in the given circle, is a fquare. Q. £. D. 132 BOOK IV. PROP. VII. PROB. BOUT a given circle I 1 to circumfcribe a fquart Draw two diameters of the given circle perpendicular to each other, and through their extremities draw 1 "> ^^^ 9 tangents to the circle ; and .Q C alio II -■ be demonftrated that that i and an d LbmmJ i s a fquare. a right angle, (B. 3. pr. 18.) = LA (conft.), ••»•- 5 in the fame manner it can »•»■ . and alfo C is a parallelogram, and becaufe they are all right angles (B. 1. pr. 34) : it is alfo evident that and " 9 "9 are equal. ,c is a fquare. Q. E. D. BOOK IV. PROP. Fill. PROB. J 33 O infcribe a circle in a given fquare. Make and draw || — and — - || (B. i. pr. 31.) and fince is a parallelogram ; = ( h yp-) is equilateral (B. 1. pr. 34.) In like manner, it can be ihown that are equilateral parallelograms ; ■■■■■«■■■■ and therefore if a circle be defcribed from the concourle of thefe lines with any one of them as radius, it will be infcribed in the given fquare. (B. 3. pr. 16.) Q^E. D. *3+ BOOK IF. PROP. IX. PROS. ]Q defer ibe a circle about a given fquare Draw the diagonals -^— — ... and — — ■ interfering each other ; then, becaufe 1 and k )ave their fides equal, and the bafe ■— — common to both, or t It (B. i.pr. 8), is bifedled : in like manner it can be mown that is bifecled ; hence \ = v = r their halves, '. ■ = — — — ; (B. i. pr. 6.) and in like manner it can be proved that If from the confluence of thefe lines with any one of them as radius, a circle be defcribed, it will circumfcribe the given fquare. Q. E. D. BOOK IF. PROP. X. PROB. O conJiruSi an ifofceles triangle, in which each of the angles at the bafe fliail n [ be double of the vertical an Take any ftraight line — and divide it fo that 4. x = (B. 2. pr. 1 1.) With ■■■■■ as radius, defcribe o and place in it from the extremity of the radius, (B. 4. pr. 1) ; draw Then \ is the required triangle. For, draw and defcribe I ) about / (B. 4. pr. 5.) .*. — - — is a tangent to I ) (B. 3. pr. 37.) = y\ (B. 3. pr. 32), 136 BOOK IF. PROP. X. PROP. add ^r to each, l! ' ▼ + W = A i B - '• P r - 5) : fince = ..... (B. 1. pr. 5.) confequently J^ = /^ -|- ^ = M^ (B. 1. pr. 32.) .*. «■■"■» = (B. 1. pr. 6.) .*. ■ — -^^— ^— iz: — — — - (conft.) .'. y\ = ▼ (B. 1. pr. 5.) =: twice x\ * 9 and confequently each angle at the bafe is double of the vertical angle. Q. E. D. BOOK IV. PROP. XL PROB. *37 N a given circle o to infcribe an equilateral and equi- angular pentagon. Conftrud: an ifofceles triangle, in which each of the angles at the bafe ihall be double of the angle at the vertex, and infcribe in the given ▲ circle a triangle equiangular to it ; (B. 4. pr. 2.) ^ and m^ ( B<I 'P r -9-) Bifedt draw and Becaufe each of the angles > +k and A are equal, the arcs upon which they ftand are equal, (B. 3. pr. 26.) and .*. i^—^— , — — ■—■ , ■ , and ■■■■»«■ which fubtend thefe arcs are equal (B.3.pr. 29.) and .*. the pentagon is equilateral, it is alfo equiangular, as each of its angles ftand upon equal arcs. (B. 3. pr. 27). Q^E. D. ■38 BOOK IF. PROP. XII. PROB. O defcribe an equilateral and equiangular penta- gon about a given circle O Draw five tangents through the vertices of the angles of any regular pentagon infcribed in the given o (B. 3. pr. 17). Thefe five tangents will form the required pentagon. Draw f— i In and (B. i.pr. 47), and — — — common ; ,7 = \A = twice and ▼ = (B. i.pr. 8.) and ^ ^ twice In the fame manner it can be demonftrated that := twice ^^ , and W = twice fe.: but = (B. 3-pr. 27), £1 BOOK IF. PROP. XII. PROB. 139 ,*, their halves = &. alfo (__ sr _J|, and ..»>•> common ; and — ■— rr — — ■— » twice — — ; In the fame manner it can be demonftrated that ^— ■---— — twice — — , In the fame manner it can be demonftrated that the other fides are equal, and therefore the pentagon is equi* lateral, it is alfo equiangular, for £^l r= twice flfct. and \^^ r= twice and therefore •'• AHw = \^B 1 m the fame manner it can be demonftrated that the other angles of the defcribed pentagon are equal. Q.E.D '1° BOOK IF. PROP. XIII. PROB. O infcribe a circle in a given equiangular and equilateral pentagon. Let tx J be a given equiangular and equilateral pentagon ; it is re- quired to infcribe a circle in it. Make y=z J^. and ^ ==" (B. i.pr. 9.) Draw Becaufe and 9 9 = - ,r=A, common to the two triangles , &c. and /. -A; Z= ••••« and =: J^ (B. I. pr. 4.) And becaufe = .*. = twice hence # rz twice is bifedted by In like manner it may be demonftrated that \^j is bifedled by ■-« « , and that the remaining angle of the polygon is bifedted in a fimilar manner. BOOK IV. PROP. XIII. PROP,. 141 Draw «^^^^— , --.----., &c. perpendicular to the lides of the pentagon. Then in the two triangles ^f and A we have ^T = mm 1 (conft.), -^^— — common, and ^^ =41 =r a right angle ; .*. — — — = .......... (B. 1. pr. 26.) In the fame way it may be mown that the five perpen- diculars on the fides of the pentagon are equal to one another. O Defcribe with any one of the perpendicu- lars as radius, and it will be the infcribed circle required. For if it does not touch the fides of the pentagon, but cut them, then a line drawn from the extremity at right angles to the diameter of a circle will fall within the circle, which has been fhown to be abfurd. (B. 3. pr. 16.) Q^E. D. H 2 BOOK IV. PROP. XIV. PROB. Bifetf: O defcribe a circle about a given equilateral and equi- angular pentagon. T and by — and -• , and from the point of fedtion, draw - := ....... (B. i. pr. 6) ; and fince in common, (B. i.pr. 4). In like manner it may be proved that =: = <— — • , and therefore nr — — — : a a 1 ••»• ti«t * Therefore if a circle be defcribed from the point where thefe five lines meet, with any one of them as a radius, it will circumfcribe the given pentagon. Q. E. I). BOOK IV PROP. XV PROP. O infcribe an equilateral and equian- gular hexagon in a given circle H3 O- From any point in the circumference of the given circle defcribe ( pamng O through its centre, and draw the diameters and draw 9 9 ......... , --..-.-- ? ......... 9 &c. and the required hexagon is infcribed in the given circle. Since paries through the centres of the circles, <£ and ^v are equilateral [ triangles, hence ^^ ' = j ^r sr one-third of two right angles; (B. i. pr. 32) but ^L m = f I 1 (B. 1. pr. 13); /. ^ = W = ^W = one-third of I I 1 (B. 1. pr. 32), and the angles vertically oppolite to theie are all equal to one another (B. 1. pr. 15), and ftand on equal arches (B. 3. pr. 26), which are fubtended by equal chords (B. 3. pr. 29) ; and fince each of the angles of the hexagon is double of the angle of an equilateral triangle, it is alfo equiangular. O E D i44 BOOK IV PROP. XVI. PROP. O infcribe an equilateral and equiangular quindecagon in a given circle. and be the fides of an equilateral pentagon infcribed in the given circle, and the fide of an inscribed equi- lateral triangle. The arc fubtended by . and _____ _6_ 1 4 of the whole circumference. The arc fubtended by 5 1 4 Their difference __: T V .*. the arc fubtended by the whole circumference. of the whole circumference. __: T V difference of Hence if firaight lines equal to ■■—.■-■■■■ be placed in the circle (B. 4. pr. 1), an equilateral and equiangular quin- decagon will be thus infcribed in the circle. Q. E. D. BOOK V. DEFINITIONS. LESS magnitude is faid to be an aliquot part or fubmultiple of a greater magnitude, when the lefs meafures the greater; that is, when the lefs is contained a certain number of times ex- actly in the greater. II. A greater magnitude is faid to be a multiple of a lefs, when the greater is meafured by the lefs ; that is, when the greater contains the lefs a certain number of times exactly. III. Ratio is the relation which one quantity bears to another of the fame kind, with refpedl to magnitude. IV. Magnitudes are faid to have a ratio to one another, when they are of the fame kind ; and the one which is not the greater can be multiplied fo as to exceed the other. The other definitions will be given throughout the book where their aid is fir ft required, v 146 AXIOMS. QUIMULTIPLES or equifubmultiples of the fame, or of equal magnitudes, are equal. If A = B, then twice A := twice B, that is, 2 A = 2 B; 3A = 3 B; 4 A = 4B; &c. &c. and 1 of A = i of B ; iofA = iofB; &c. &c. II. A multiple of a greater magnitude is greater than the fame multiple of a lefs. Let A C B, then 2AC2B; 3 ACZ3B; 4 A C 4 B; &c. &c. III. That magnitude, of which a multiple is greater than the fame multiple of another, is greater than the other. Let 2 A m 2 B, then ACZB; or, let 3 A C 3 B, then ACZB; or, let m A C m B, then ACB. BOOK V. PROP. I. THEOR. i*7 F any number of magnitudes be equimultiples of as many others, each of each : what multiple soever any one of the fir Jl is of its part, the fame multiple Jhall of the fir Jl magnitudes taken together be of all the others taken together. LetQQQQQ be the fame multiple of Q, that WJFW is of f . that OOOOO « of O. Then is evident that QQQQQ1 [Q is the fame multiple of 4 OQOOQ [Q which that QQQQQ isofQ ; becaufe there are as many magnitudes in 4 QQQQQ fffff > L OOOOO V o as there are in QQQQQ = Q . The fame demonftration holds in any number of mag- nitudes, which has here been applied to three. .*. If any number of magnitudes, &c. 1 48 BOOK V. PROP. II. THEOR. F the jirjl magnitude be the fame multiple of the fecond that the third is of the fourth, and the fifth the fame multiple of the fecond that the fix th is oj the fourth, then foall the firjl, together with the fifth, be the fame multiple of the fecond that the third, together with the fixth, is of the fourth. Let \ , the firft, be the fame multiple of ) , the fecond, that O0>O> tne tnu 'd> is of <j>, the fourth; and let 00^^, the fifth, be the fame multiple of ) , the fecond, that OOOOj l ^ e ^ xtn > 1S °f 0>> l ^ e fourth. Then it is evident, that J > , the firft and fifth together, is the fame multiple of , the fecond, that l \ \, the third and fixth together, is of looooj the fame multiple of (J> , the fourth ; becaufe there are as many magnitudes in -j _ z= as there are m looooj - ° ■ /. If the firft magnitude, &c. BOOK V. PROP. III. THEOR. 149 F the jirjl of four magnitudes be the fame multiple of the fecond that the third is of the fourth, and if any equimultiples whatever of the fir ft and third be taken, thofe Jliall be equimultiples ; one of the fecond, and the other of the fourth. The First. The Second. Let -i take \ • be the lame multiple of The Third. The Fourth. which J I is of A ; y the fame multiple of < ♦ ♦♦♦ which <; is of ♦ ♦' that <! Then it is evident, The Second. ► is the fame multiple of | i jo BOOK V. PROP. III. THEOR. ♦ ♦♦♦ which < ♦♦♦♦ ♦♦♦♦ The Fourth. • is of A ; becaufe < > contains < > contains as many times as y contains ♦ ♦ ♦ ♦ > contains ^ ♦♦♦♦ ♦♦♦♦ ♦♦♦♦ The fame reafoning is applicable in all cafes. .'. If the firft four, &c. BOOK V. DEFINITION V. '5 1 DEFINITION V. Four magnitudes, £», , ^ , ^, are laid to he propor- tionals when every equimultiple of the firft and third be taken, and every equimultiple of the fecond and fourth, as, of the firft &c. of the fecond of the third + ^ ♦♦♦ ♦ ♦♦♦ ♦ ♦♦♦♦ ♦♦♦♦♦♦ &c. of the fourth If < &c. &c. Then taking every pair of equimultiples of the firft and third, and every pair of equimultiples of the fecond and fourth, = °rZ, ■■ = o rZ| SOT" 3 : or ^ : or ^ ;, = or 3 :. = or 3 ;, = or 3 !» — or ~l ♦ ♦ ♦ ♦ then will ^ ^ ♦ ♦ I 5 2 BOOK V. DEFINITION V. That is, if twice the firft be greater, equal, or lefs than twice the fecond, twice the third will be greater, equal, or lefs than twice the fourth ; or, if twice the firft be greater, equal, or lefs than three times the fecond, twice the third will be greater, equal, or lefs than three times the fourth, and so on, as above exprelfed. in then will • •• c, = or Zl • •• c, = or Zl • #• c, ^ or Z3 • •• d, = or n ••• 1=, = or Z] &c. [♦♦♦ c=, = or Zl ♦♦♦ c, = or Zl - ♦♦♦ c, = or ^ ♦ ♦♦ & = or z: ,♦♦♦ c = or Zl &c. &c. &c. In other terms, if three times the firft be greater, equal, or lefs than twice the fecond, three times the third will be greater, equal, or lefs than twice the fourth ; or, if three times the firft be greater, equal, or lefs than three times the fecond, then will three times the third be greater, equal, or lefs than three times the fourth ; or if three times the firft be greater, equal, or lefs than four times the fecond, then will three times the third be greater, equal, or lefs than four times the fourth, and so on. Again, BOOK V. DEFINITION V. J 53 If < then will tiff cz> ™ ^™ or or ^1 •••# c __ or ZJ •••• cz, __ or Z] •••• c :m or Z] &c. ♦♦♦♦ IZ, ^^ or Zl ♦ ♦♦♦ I— 9 = or Z] •♦♦♦♦ L— > — or Z] ♦♦♦♦ L-~ 9 = or Zl [♦♦♦♦ c, — or Z3 &c. &c. And so on, with any other equimultiples of the four magnitudes, taken in the fame manner. Euclid exprefles this definition as follows : — The firft of four magnitudes is faid to have the fame ratio to the fecond, which the third has to the fourth, when any equimultiples whatfoever of the firft and third being taken, and any equimultiples whatfoever of the fecond and fourth ; if the multiple of the firft be lefs than that of the fecond, the multiple of the third is alfo lefs than that of the fourth ; or, it the multiple of the firft be equal to that of the fecond, the multiple of the third is alfo equal to that of the fourth ; or, ir the multiple of the firft be greater than that of the fecond, the multiple of the third is alfo greater than that of the fourth. In future we fhall exprefs this definition generally, thus : If M # C, = or Zl m |, when M ▲ CZ, = or "1 w ^ 154 BOOK V. DEFINITION V. Then we infer that % , the firft, has the fame ratio to | , the fecond, which ^, the third, has to ^P the fourth : expreffed in the fucceeding demonstrations thus : • :■ :: ♦: V; or thus, # : = ♦ : 9 9 or thus, — = — - : and is read, V " as £ is to , so is ^ to ^. And if # : :: ^ : f we mall infer if M § C, =: or ^] //; , then will M ^ C = or Z3 /« ^. That is, if the firft be to the fecond, as the third is to the fourth ; then if M times the firft be greater than, equal to, or lefs than tn times the fecond, then fhall M times the third be greater than, equal to, or lefs than m times the fourth, in which M and m are not to be confidered parti- cular multiples, but every pair of multiples whatever; nor are fuch marks as Q, ^, , &c. to be confidered any more than reprefentatives of geometrical magnitudes. The ftudent fhould thoroughly underftand this definition before proceeding further. BOOK V. PROP. IV. THEOR. 155 F the fir jl of four magnitudes have the fame ratio to the fecond, which the third has to the fourth, then any equimultiples whatever of the firfi and third shall have the fame ratio to any equimultiples of the fecond and fourth ; viz., the equimultiple of the firfl fliall have the fame ratio to that of the fecond, which the equi- multiple of the third has to that of the fourth. Let :>:.*♦ :^, then 3 :2|::34:2f, every equimultiple of 3 and 3 ^ are equimultiples of and ^ , and every equimultiple of 2 | | and 2 JP , are equimultiples of 1 1 and ^ (B. 5, pr. 3.) That is, M times 3 and M times 3 ^ are equimulti- ples of and ^ , and m times 2 1 1 and m 2 S are equi- multiples of 2 I I and 2 ^ • but • I I • • ^ • W (hyp); .*. if M 3 EZ, =, or —j «/ 2 |, then M 3 ^ CZ . =, or ^ ;« 2 f (def. 5.) and therefore 3 : z | | :: 3 ♦ ; 2 ^ (def. 5.) The fame reafoning holds good if any other equimul- tiple of the firft and third be taken, any other equimultiple of the fecond and fourth. .*. If the firft four magnitudes, &c. i 5 6 BOOK V. PROP. V. THEOR. F one magnitude be the fame multiple of another, which a magnitude taken from thefirjl is of a mag- nitude taken from the other, the remainder Jhall be the fame multiple of the remainder, that the whole is of the whole. Q Let OQ = M ' D and = M'., o <^>Q> minus = M' minus M' ■, O /. & = M' (* minus ■), and .*. Jp^ =M' A. ,*. If one magnitude, &c. BOOK V. PROP. VI. THEOR. 157 IBg<BI Km /*llo * ■P* 1 g\y/^a Mm Hr '-s ^V t Vara F /wo magnitudes be equimultiples of two others, and if equimultiples of t lief e be taken from the fir ft two, the remainders are either equal to thefe others, or equimultiples of them. Q Let = M' ■ ; and QQ = M' a ; o then minus m m = M' * minus m m = (M' minus /»') ■ , ar >d OO mmus w ' A = M' a minus m a = (M' minus /»') a • Hence, (M' minus tri) ■ and (M' minus rri) a are equi- multiples of ■ and a , and equal to ■ and a , when M' minus m sr 1 . .'. If two magnitudes be equimultiples, &c. i 5 8 BOOK V. PROP. A. THEOR. F the fir Jl of the four magnitudes has the fame ratio to the fecond which the third has to the fourth, then if the firfi be greater than the fecond, the BfeSSi] third is a/fo greater than the fourth ; and if equal, equal; ij fiefs, lefs. Let £ : | | : : qp : ; therefore, by the fifth defini- tion, if |f C H, then will f f C but if # EI ■, then ## [= ■■ and ^ CO, and .*. ^ C ► • Similarly, if £ z=, or ^] ||, then will f =, or ^| ► . .*. If the firft of four, &c. DEFINITION XIV. Geometricians make ufe of the technical term " Inver- tendo," by inverfion, when there are four proportionals, and it is inferred, that the fecond is to the firft: as the fourth to the third. Let \ : B : : C : D , then, by " invertendo" it is inferred B : A :: U : C. BOOK V. PROP. B. THEOR. '59 F four magnitudes are proportionals, they are pro- portionals alfo when taken inverfely. Let ^ : Q : : ■ : { ► , then, inverfely, Q:f :: : ■ . If M qp ID ot Q, then M|Uw by the fifth definition. Let M ■ Zl ^ O, that is,ffl[jCMf , ,'. M 1 H ;» , or, /» EM|; .*. iffflQCMf , then will m EM| In the fame manner it may be mown, that if m Q = or Z3 M ^ , then will /» ;=, or 13 M | | ; and therefore, by the fifth definition, we infer that Q : ^ : # : ■. .*. If four magnitudes, &c. 160 ROOKV. PROP. C. THEOR. F the jirjl he the fame multiple of the fecond, or the fame part of it, that the third is of the fourth ; the firjl is to the fecond, as the third is to the fourth. Let _ _ , the firft,be the fame multiple of Q, the fecond, that , the third, is of A, the fourth. Then ■■ : * :: il : * ♦ ♦' becaufe J is the fame multiple of that is of Wk (according to the hypothcfis) ; ■ ■ • ■■ and M - ; is taken the fame multiple of" that M is of J , .*. (according to the third propofition), M _ is the fame multiple of £ that M is of £ . BOOK V. PROP. C. THEOR. 161 Therefore, if M . be of £ a greater multiple than m £ is, then M is a greater multiple of £ tnan m £ is ; that is, if M 5 \ be greater than w 0, then M will be greater than m ^ ; in the fame manner it can be fhewn, if M ! be equal m Q. then M will be equal ;« £. And, generally, if M f CZ, = or ZD m £ then M will be CZ, = or ^ m 6 ; .*. by the fifth definition, ■ ■•'♦♦••• ■ ■ Next, let be the fame part of ! that 4k is of r . In this cafe alfo : j :: A : T. For, becaufe A is the fame part of ! ! that A is of ■ ■ ♦ ♦ 1 62 BOOK V. PROP. C. THEOR. therefore J . is the fame multiple of that is of £ . Therefore, by the preceding cafe, ■ ■ . a •• • ▲ • ■■'•"♦♦ "■• and .*. £ : . . :: £ : . , by proportion B. /. If the firft be the fame multiple, &c. BOOK V. PROP. D. THEOR. 163 the fit -ft be to the fecond as the third to the fourth, and if the Jirji be a multiple, or a part of the fecond ; the third is the fame multiple, or the fame part of the fourth. L >•• ■ and firft, let •V je a multiple | |. (hall b e the fame multiple of ■■ . First. • Second. ■ Third. Fourth. ♦ ♦ w O QQ QQ OO Take a QQ _ • Whatever multiple : ^L isofH take OO OO the fam< ; multiple of ■ , then, becaufe and of the fecond and fourth, we have taken equimultiples, and yT/C> therefore (B. 5. pr. 4), 1 64 BOOK V. PROP. D. THEOR. : QQ :: JJ : OO' but(C0nft)> -QQ ••( B '5F-A-)^ 4 - oc and /Ty\ is the fame multiple of ^ that is of ||. Next, Id | : : : JP : £, and alfo | | a part of ; then <9 mail be the fame part of ^ . nverfely (B •5-). ••" -..♦♦ ■"♦♦ but | is a part .*. that is, •i is a multiple of | | ; ♦♦ ic fr\** lorviP i-v^ ii /. by the preceding cafe, . is the fame multiple of that is, ^ is the fame part of , that | | is of . .*. If the firft be to the fecond, &c. BOOK V. PROP. VII. THEOR 165 QUAL magnitudes have the fame ratio to the fame magnitude t and the fame has the fame ratio to equal magnitudes. Let $ = 4 and any other magnitude ; then # : = + : and : # = : 4 Becaufe £ = ^ , .-. M • = M 4 ; .\ if M # CZ, = or ^ w , then M + C, = or 31 m I, and .-. • : I = ^ : | (B. 5. def. 5). From the foregoing reafoning it is evident that, if m C> = or ^ M 0, then m C = or Zl M ^ /.■•=■ 4 (B. 5. def. 5). /. Equal magnitudes, &c. 1 66 ROOK V. DEFINITION VII. DEFINITION VII. When of the equimultiples of four magnitudes (taken as in the fifth definition), the multiple of the firfl: is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth ; then the firfl is laid to have to the fecond a greater ratio than the third magnitude has to the fourth : and, on the contrary, the third is laid to have to the fourth a lefs ratio than the firfl: has to the fecond. If, among the equimultiples of four magnitudes, com- pared as in the fifth definition, we fhould find • ####[Z ,but + ♦ ♦ ♦ ♦ s or Zl ffff,orifwe fhould find any particular multiple M' of the firfl: and third, and a particular multiple tri of the fecond and fourth, fuch, that M' times the firfl: is C tri times the fecond, but M' times the third is not [Z tri times the fourth, /. e. = or ~1 tri times the fourth ; then the firfl is faid to have to the fecond a greater ratio than the third has to the fourth ; or the third has to the fourth, under fuch circumftances, a lefs ratio than the firfl has to the fecond : although feveral other equimultiples may tend to fhow that the four mag- nitudes are proportionals. This definition will in future be exprefled thus : — If M' ^ CI tri O, but M' 1 = or Z tri ► , then ^P : Q rZ H : ► • In the above general exprefllon, M' and tri are to be confidered particular multiples, not like the multiples M BOOK V. DEFINITION VII. 167 and m introduced in the fifth definition, which are in that definition confidered to be every pair of multiples that can be taken. It muff, alio be here obferved, that ^P , £~J, 1 1 , and the like fymbols are to be confidered merely the repre- fentatives of geometrical magnitudes. In a partial arithmetical way, this may be fet forth as follows : Let us take the four numbers, : , 7, i;, and Firft. Second. Third. Fourth. 8 7 10 9 16 H 20 I O 24 21 3° 27 32 28 40 36 40 35 5° 45 48 42 60 54 56 49 7° 6 3 64 5° 80 72 72 63 90 8: 80 70 100 90 88 V no 99 96 84 120 108 104 9 1 '3° 117 112 98 j 40 126 &C. &c. &c Sec. Among the above multiples we find r C 14 and z tZ that is, twice the firft is greater than twice the lecond, and twice the third is greater than twice the fourth; and i 6 ^ 2 1 and 2 ^3 that is, twice the firft is lefs than three times the fecond, and twice the third is lefs than three times the fourth ; and among the fame multiples we can find Hi 56 and v IZ that is, 9 times the firft is greater than 8 times the fecond, and 9 times the third is greater than 8 times the fourth. Many other equimul- 1 68 BOOK V. DEFINITION VII. tiples might be selected, which would tend to fliow that the numbers ?, 7, 10, were proportionals, but they are not, for we can find a multiple of the firft: £Z a multiple of the fecond, but the fame multiple of the third that has been taken of the firft: not [Z the fame multiple of the fourth which has been taken of the fecond; for inftance, 9 times the firft: is Q 10 times the fecond, but 9 times the third is not CI I0 times the fourth, that is, 72 EZ 70, but 90 not C or 8 times the firft: we find C 9 times the fecond, but 8 times the third is not greater than 9 times the fourth, that is, 64 C 63, but So is not C When any fuch multiples as thefe can be found, the firft: ( !)is faid to have to the fecond (7) a greater ratio than the third (10) has to the fourth and on the contrary the third (10) is faid to have to the fourth a lefs ratio than the firft: 3) has to the fecond (7). BOOK V. PROP. VIII. THEOR. 169 F unequal magnitudes the greater has a greater ratio to the fame than the lefs has : and the fame magnitude has a greater ? atio to the lefs than it has to the greater. Let I I and be two unequal magnitudes, and £ any other. We mail firft prove that | | which is the greater of the two unequal magnitudes, has a greater ratio to £ than |, the lefs, has to A j that is, ■ : £ CZ r : # ; A take M' 1 1 , /»' , M' , and m ; fuch, that M' a and M' g| mail be each C ; alfo take m £ the lean: multiple of £ , which will make m' M' =M' .*. M' is not ;;/ but M' I I is |~ m £ , for, as m' is the firft multiple which fir ft becomes CZ M'|| , than (m minus 1) orw' ^ minus Q is not I M' 1 1 . and % is not C M' A, /. tri minus that + muft be Z2 M' | + M' a ; A is, m muft be — 1 M' .'. M' I I is C *»' j but it has been ftiown above that z 170 BOOK V. PROP. VIII. THEOR. M' is not C»'§, therefore, by the feventh definition, A | has to £ a greater ratio than : . Next we mall prove that £ has a greater ratio to , the lefs, than it has to , the greater; or, % : I c # : ■• Take m £ , M' , ni %, and M' |, the fame as in the firff. cafe, fuch, that M' a and M' | | will be each CZ > ar >d f» % the leaft multiple of £ , which firfr. becomes greater than M' p = M' ■ . .". ml % minus £ is not d M' j | , and f is not C M' ▲ ; confequently ot' minus # + # is Zl M' | + M' ▲ ; .'. »z' is ^ M' | | , and .*. by the feventh definition, A has to a greater ratio than Q has to || . .*. Of unequal magnitudes, &c. The contrivance employed in this proportion for finding among the multiples taken, as in the fifth definition, a mul- tiple of the firft greater than the multiple of the fecond, but the fame multiple of the third which has been taken of the firft, not greater than the fame multiple of the fourth which has been taken of the fecond, may be illuftrated numerically as follows : — The number 9 has a greater ratio to 7 than has to 7 : that is, 9 : 7 CI : 7 5 or, b -}- 1 : 7 fZ - '-7- BOOK V. PROP. Fill. THEOR, 171 The multiple of 1, which firft becomes greater than 7, is 8 times, therefore we may multiply the firft and third by 8, 9, 10, or any other greater number ; in this cafe, let us multiply the firft and third by 8, and we have 64^-8 and : again, the firft multiple of 7 which becomes greater than 64 is 10 times; then, by multiplying the fecond and fourth by 10, we fhall have 70 and 70 ; then, arranging thefe multiples, we have — 8 times 10 times 8 times 10 times the first. the second. the third. the fourth. 64+ 8 70 70 Confequently , «-|- 8, or 72, is greater than - : , but is not greater than 70, .\ by the feventh definition, 9 has a greater ratio to 7 than has to - . The above is merely illuftrative of the foregoing demon- ftration, for this property could be fhown of thefe or other numbers very readily in the following manner ; becaufe, if an antecedent contains its confequent a greater number of times than another antecedent contains its confequent, or when a fraction is formed of an antecedent for the nu- merator, and its confequent for the denominator be greater than another fraction which is formed of another antece- dent for the numerator and its confequent for the denomi- nator, the ratio of the firft antecedent to its confequent is greater than the ratio of the laft antecedent to its confe- quent. Thus, the number 9 has a greater ratio to 7, than 8 has to 7, for - is greater than -. Again, 17 : 19 is a greater ratio than 13:15, becaufe 17 17 X 15 25,5 , 13 13 X 19 247 , 5 - ^T>TTi - isi' and I5 = T^T 9 = «? hence « IS evident that ?|f is greater than ~ t .-. - is greater than 172 BOOK V. PROP. VIII. THEOR. — , and, according to what has been above fhown, \j has to 19 a greater ratio than 13 has to 15. So that the general terms upon which a greater, equal, or lefs ratio exifts are as follows : — A C If -g be greater than ■=-, A is faid to have to B a greater A C ratio than C has to D ; if -^ be equal to jt, then A has to B the fame ratio which C has to D ; and if ^ be lefs than c ^, A is faid to have to B a lefs ratio than C has to D. The ftudent mould underftand all up to this propofition perfectly before proceeding further, in order fully to com- prehend the following propofitions of this book. We there- fore ftrongly recommend the learner to commence again, and read up to this {lowly, and carefully reafon at each flep, as he proceeds, particularly guarding againft the mifchiev- ous fyflem of depending wholly on the memory. By fol- lowing thefe inftruclions, he will find that the parts which ufually prefent confiderable difficulties will prefent no diffi- culties whatever, in profecuting the ftudy of this important book. BOOK V. PROP. IX. THEOR. 173 AGNITUDES which have the fame ratio to the fame magnitude are equal to one another ; and thofe to which the fame magnitude has the fame ratio are equal to one another. Let ▲ : I I : : £ : 1 1, then ^ =f . For, if not, let ▲ C • > then will ♦ : € C # : ■ (B. 5- pr- 8), which is abfurd according to the hypothecs. .*. ^ is not C % ' In the fame manner it may be mown, that £ is not CZ t ' Again, let | : ▲ : : # ? then will ^ = . For (invert.) + : - # • | ? therefore, by the firfl cafe, ▲ =0. .*. Magnitudes which have the fame ratio, 6cc. This may be fhown otherwife, as follows : — Let \ : B ZZZ ' : C> then Br:C, for, as the fraction — = the fraction — , and the numerator of one equal to the B c * numerator of the other, therefore the denominator of thefe fractions are equal, that is BrC. Again, if B : = C : A, B = C. For, as - = ^, B muft = C- *74 BOOK V. PROP. X. THEOR. HAT magnitude which has a greater ratio than another has unto the fame magnitude, is the greater of the two : and that magnitude to which the fame has a greater ratio than it has unto another mag- nitude, is the lefs of the two. Let jp : C # : 1 1, then ^ C # • For if not, let W — or ~l ^ ; then, qp : = # : ( B - 5- P r - 7) or ^ : 1 13 9 : (B. 5. pr. 8) and (invert.), which is abfurd according to the hypothecs. .*. ^p is not = or ^ £ , and .*. ^ muftbe CZ •• Again, let ? : # C ! : JP, then, ^ H V- For if not, £ muft be C or = ^ , then |:|^ : JP (B. 5. pr. 8) and (invert.) ; == I : ■ (B. 5. pr. 7), which is abfurd (hyp.); /. £ is not CZ or = ^P, and .*. A muft be 13 •• or .*. That magnitude which has, 6cc. BOOK V. PROP. XL THEOR. l 75 ATI OS that are the fame to the fame ratio, are the fame to each other. Let ♦ : ■ r= % : and : = A : •, then will ^ : | | = A : •. For if M # Cf => or 13 » ■ , then M £ C» =» or 3 z» p , and if M C =:, or ^ /« p , then M A CZ, :=, or ^ m •, (B. 5. def. 5) ; \ if M ♦ C, =, or 33 m ■ , M A CZ, =, or 3 w • . and .*. (B. 5. def. 5) + : B = A : •• .*. Ratios that are the fame, &c. i 7 6 BOOK V. PROP. XII. THEOR. F any number of magnitudes be proportionals, as one of the antecedents is to its confequent, Jo Jhall all the antecedents taken together be to all the confequents. Let H : • = U : O = ► : ' = •:▼ = *:•; then will | | : £ ss ■ +D + +• + *:# + <>+ +▼ + •• For ifM|C m % , then M Q [Z m £>, and M . C m M • C m ▼ , alfo MaC« •• (B. 5. def. 5.) Therefore, if M | | CZ m , then will M|+MQ + M +M. + Mi, or M J| + O + + • + A ) be grater than m £ 4" w C 4" m "f" m T "I" w •' or^«(#+0+ +▼+•)■ In the fame way it may be mown, if M times one of the antecedents be equal to or lefs than m times one of the con- fequents, M times all the antecedents taken together, will be equal to or lefs than m times all the confequents taken together. Therefore, by the fifth definition, as one of the antecedents is to its confequent, fo are all the antecedents taken together to all the confequents taken together. .*. If any number of magnitudes, &c. BOOK V. PROP. XIII. THEOR. [ 77 F the jirji has to the fecond the fame ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth ; the firjijhall alfo have to the fecond a greater ratio than the fifth to the fixth. Let 9 : Q = ■ : >, but ■ : C O '- •> then f:OCO:l For, becaufe | | : CO : i) t ^ iere are *° me mu ^" tiples (M' and ni) of j | and <^, and of and £ . fuch that M' | CZ ni , but M' <^ not C ni £, by the feventh definition. Let thefe multiples be taken, and take the fame multiples of fM and f^. /. (B. 5. def. 5.) if M' 9 C, =, or Z\ ni Q ; then will M' ■ IZ, =, or ^2 m ' , but M' I C m ' (connruclion) ; .-. m ' qp tz ni Q, but M' <^> is not CZ ni £ (conftrudtion) ; and therefore by the feventh definition, .*. If the firft has to the fecond, &c. A A i/8 BOOK V. PROP. XIV. THEOR. F the firji has the fame ratio to the fecond which the third has to the fourth ; then, if the fir j} be greater than the third, the fecond foall be greater than the fourth; and if equal, equal ; and if lefs, lefs. Let ^ : Q : : : + , and firft fuppofe V CZ |, then will O CZ #. For f : O C I U (B. 5- pr- 8). and by the hypothefis, ^ I Q = : + ; .*.■:♦ CB:D(B. 5 .pr. i3). .*. ♦ Zl D (B. S- pr. io.), orOCf Secondly, let ■ = , then will ^J zz 4 . For * : (J = : Q (B. 5 . pr. 7 ), and fl : Q = 9 : ^ (hyp.) ; .\ ■ :Q= M :♦ (B. 5 . P r. n), and ,\ Q = + (B. 5, pr. 9). Thirdly, if JP Z] , then will O Z] ♦ ; becaufe | CI ^ and : + = ^ : O ; .*. ♦ c O? by tne ^ft ca ^" e » that is, Q Zl ♦ • '. If the firft has the fame ratio, &c. BOOK V. PROP. XV. THEOR. 179 A.GNITUDES have the fame ratio to one another which their equimultiples have. Let £ and be two magnitudes ; then, # : ft :: M' % : M' I. For • :■ .*. • "• I "4 • : 4 • (B. 5- P r - I2 )- And as the fame reafoning is generally applicable, we have • : ■ : : M' A : M' ■ . .*. Magnitudes have the fame ratio, &,c. 180 BOOK V. DEFINITION XIII. DEFINITION XIII. The technical term permutando, or alternando, by permu- tation or alternately, is ufed when there are four propor- tionals, and it is inferred that the firfl has the fame ratio to the third which the fecond has to the fourth ; or that the tirft is to the third as the fecond is to the fourth : as is ihown in the following propofition : — Let : + ::?:■' by '* permutando" or "alternando" it is inferred . : ^ :: ^ : |. It may be neceffary here to remark that the magnitudes , A, M, ||, muft be homogeneous, that is, of the fame nature or fimilitude of kind ; we muft therefore, in fuch cafes, compare lines with lines, furfaces with furfaces, folids with folids, &c. Hence the ftudent will readily perceive that a line and a furface, a furface and a folid, or other heterogenous magnitudes, can never ftand in the re- lation of antecedent and confequent. BOOK V. PROP. XVI. THEOR. 81 F four magnitudes of the fame kind be proportionals, they are alfo proportionals when taken alternately. Let <|p : Q :: : 4 , then ::0'#. ForM 9 : M Q :: * : O ( B - 5- P r - I 5)> and M|:MQ:: : + (hyp.) and (B. 5. pr. 1 1 ) ; alfo m : ;;/ ^ :: : ▲ (B. 5. pr. 15); ,\ M qp : M Q :: « : ;» 4 (B. 5. pr. 14), and .*. if M ^ C» =» or I] w | , then will M Q d, => or ^ /// ^ (B. 5. pr. 14) ; therefore, by the fifth definition, v- m o: ♦• .*. If four magnitudes of the fame kind, &c. 1 82 BOOK V. DEFINITION XVI. DEFINITION XVI. Dividendo, by divifion, when there are four proportionals, and it is inferred, that the excefs of the firfr. above the fecond is to the fecond, as the excefs of the third above the fourth, is to the fourth. Let A : B : : C : D ; by " dividendo " it is inferred A minus B : B : : i minus ) : D. According to the above, A is fuppofed to be greater than B, and C greater than ; if this be not the cafe, but to have B greater than A, and greater than C > B and can be made to ftand as antecedents, and A and C as confequents, by " invertion " B : A : •. D : C ; then, by "dividendo," we infer B minus A : A : : minus C : C • BOOK V. PROP. XVII. THEOR. 183 jF magnitudes, taken jointly, be proportionals, they fliall alfo be proportionals when taken feparately : that is, if two magnitudes together have to one of them the fame ratio which two others have to one of thefe, the remaining one of the fir ft two foall have to the other the fame ratio which the remaining one of the lafi two has to the other of thefe. Let f + O: Q:: : + ♦ : ♦, then will :Q:: % : ♦■ Take M V IZ m Q to each add M Q, then we have M ^ + M U[Z>«U + M Q ? orM(^ + 0) c (« + M)Q: but becaufe ^P + Q:Q::B + 4:^ (hyp.), andM(^P + 0)EZ(* + M)D; .*. M( +^)CI( W + M)4 (B. 5. def. 5 ); /. M ■ + M + C m + + M # ; .*. M i tZ w ♦ , by taking M + from both fides : that is, when MfC* O, then M Cw^, In the fame manner it may be proved, that if M ^P = or ^ /« Q, then will M = or ^ « 4 and /. ^ : Q : : > : 4 (B. 5. def. 5). .*. If magnitudes taken jointly, &c. l8 4 BOOK V. DEFINITION XV. DEFINITION XV. The term componendo, by compofition, is ufed when there are four proportionals ; and it is inferred that the firft toge- ther with the fecond is to the fecond as the third together with the fourth is to the fourth. Let A : B : : : D ; then, by the term " componendo," it is inferred that A + B : B :: + D : D. By " invertion" B and p may become the firlt and third, A and the fecond and fourth, as B : A : : D : , then, by " componendo," we infer that B + A : A : : D + : . BOOK V. PROP. XVIII. THEOR. 185 F magnitudes, taken feparately, be proportionals, they Jhall alfo be proportionals when taken jointly : that is, if the firjl be to the fecond as the third is to the fourth, the firjl and fecond together Jliall be to the fecond as the third and fourth together is to the fourth. Let * : O then * + Q : Q for if not, let ^ -{- Q fuppofing Q .'. W : O ' = but ;■ + ♦:♦; not = ^ ; • (B. 5. pr. 17); -.Q:: : ♦ (hyp.); : :: I : 4 ( B - 5- P r - JI ); ••••=♦ (B- 5- P^ 9). which is contrary to the fuppofition ; .*. £ is not unequal to ^ ; that is =1 4 5 '. If magnitudes, taken feparately, &c. B B i86 BOOK V. PROP. XIX. THEOR. F a whole magnitude be to a whole, as a magnitude taken from the firft, is to a magnitude taken from the other ; the remainder Jhall be to the remainder, as the whole to the whole. then will Q : :: ^ + D :| + », .\ G : again Q : but * + O therefore ^J ". If a whole magnitude be to a whole, &c. *■■ -:■(<! ivid.), :: W :■ (alter.), :■+ # .'V : ■ hyp.) ♦ "* + U ■ +♦ (B. 5. pr. 11). DEFINITION XVII. The term " convertendo," by converfion, is made ufe of by geometricians, when there are four proportionals, and it is inferred, that the firft. is to its excefs above the fecond, as the third is to its excefs above the fourth. See the fol- lowing propofition : — BOOK V. PROP. E. THEOR. 187 F four magnitudes be proportionals, they are alfo proportionals by converjion : that is, the fir Jl is to its excefs above the fecond, as the third to its ex- cefs above the fourth. then lhall # O : • • : ■ : ■ > Becaufe therefore 1 .-. o 10:0: |:0"B • ♦ • ::■♦ !♦♦; (divid.), I (inver.), I (compo.). ,*. If four magnitudes, &c. DEFINITION XVIII. " Ex aBquali" (fc. diflantia), or ex zequo, from equality of diftance : when there is any number of magnitudes more than two, and as many others, fuch that they are propor- tionals when taken two and two of each rank, and it is inferred that the nrft is to the laft of the firft rank of mag- nitudes, as the firft is to the laft of the others : " of this there are the two following kinds, which arife from the different order in which the magnitudes are taken, two and two." 188 BOOK V. DEFINITION XIX. DEFINITION XIX. " Ex aequali," from equality. This term is ufed amply by itfelf, when the firft magnitude is to the fecond of the firft rank, as the nrft to the fecond of the other rank ; and as the fecond is to the third of the hrft rank, fo is the fecond to the third of the other ; and fo on in order : and the in- ference is as mentioned in the preceding definition ; whence this is called ordinate proportion. It is demonftrated in Book 5. pr. 22. Thus, if there be two ranks of magnitudes, A, B, 1. , P, E, F, the nrft rank, and L, M, , , P, Q, the fecond, fuch that A : B : : L : M, B : ( :: M : , : : I : : : , : E : : : P, E : F : : P : Q ; we infer by the term " ex asquali" that A : F :: L : Q. BOOKV. DEFINITION XX. 189 DEFINITION XX. " Ex squali in proportione perturbata feu inordinata," from equality in perturbate, or diforderly proportion. This term is ufed when the firft magnitude is to the fecond of the firft rank as the laft but one is to the laft of the fecond rank ; and as the fecond is to the third of the firft rank, fo is the laft but two to the laft but one of the fecond rank ; and as the third is to the fourth of the firft rank, fo is the third from the laft to the laft but two of the fecond rank ; and fo on in a crofs order : and the inference is in the 1 8th definition. It is demonstrated in B. 5. pr. 23. Thus, if there be two ranks of magnitudes, A, B, C, D, , , the firft rank, and , , N , O , P , Q , the fecond, fuch that A:B::P:Q,B:C::0:P, C : D :: N : O, D : :: : N, i : F :: : I ; the term " ex a?quali in proportione perturbata feu inordi- nata" infers that A : :: : Q. 190 BOOK V. PROP. XX. THEOR. F there be three magnitudes, and other three, which, taken two and two, have the fame ratio ; then, if' the Jirjl be greater than the third, the fourth Jha I I be greater than the fixth ; and if equal, equal ; and if lefs, lefs. Let ^P, {^J, ||, be the fir ft three magnitudes, and ^, (3, ( , be the other three, fuch that V :0 ::+ : C> , an <l O : M '-'-O '■ O • Then, if ^ d» => or ^ , then will ^ C ==, orZ3 t From the hypothecs, by alternando, we have andQ :0 ::■ :•; /. ^ : ♦ ::| | : t (B. 5 . pr. n); •\ if f C» =. or D , tlien will + C =, or3 # (B. 5. pr. 14). .*. If there be three magnitudes, 6cc. BOOK V. PROP. XXI. THEOR. 191 F there be three magnitudes, and other three which have the fame ratio, taken two and two, but in a crofs order ; then if the firjl magnitude be greater than the third, the fourth Jhall be greater than the jixth ; and if equal, equal ; and if lefs, lefs. Let p, £ , ||, be the firft three magnitudes, and ^, 0>> ( ? the other three, fuch that \ : £ :: O ••> and £ : ■ :: ♦ : 0>« Then, if I C =, or Z2 I will ♦[=,=,=! |. then Firft, let < be CI ■ : then, becaufe £ is any other magnitude, f :iC|:i (B. 5. pr. 8); butO M :: :4 (hyp-); .*. 0> = (B. 5. pr. 13); and becaufe jfe : ■ :: ^ : O ( n yp-) 5 .*•■ :A -O :♦ (in*.). and it was fhown that £ : C | '• A < .'. O : < C O =♦ (B. s-pr. 13); 1 92 BOOK V. PROP. XXI. THEOR. •• • =] ♦, that is ^ CI | . Secondly, let = | | ; then {hall ^ = ) . For becaufe — |, * : * = ■ : A (B. 5-F- 7); but : A = O : 1 (hyp.), and I I : 4b = O • ^ (hyp- and inv.), .-. O : # = : ♦ (B. 5. P r. 11), •'• ♦ = • (B- 5- P^ 9)- Next, let be Z2 ■• then ^ fhall be ^ ; for|C ', and it has been fhown that | : 4fc = : $, and ^ : ' s = ; 1 : O; /. by the firft cafe is CZ ^j that is, ^ ^ ). /. If there be three, &c. BOOK V. PROP. XXII. THEOR. *93 F there be any number of magnitudes, and as many others, which, taken two and two in order, have the fame ratio ; the firft Jhall have to the laji of the fir Jl magnitudes the fame ratio which the fir /I of the others has to the laji of the fame. N.B. — This is ufually cited by the words "ex trqua/i," or "ex aquo." Firft, let there be magnitudes f^ , + , 1 1 , and as many others ▲ ,(^, ) , luch that V '•♦ "♦ -O* and^ : | :: <^> * I ; then mail ^ : { : : ^ • O . Let thefe magnitudes, as well as any equimultiples whatever of the antecedents and confequents of the ratios, ftand as follows : — and becaufe qp : ^ : : ^ : 0> » .\ M fp : »i + : : M ^ : /» £> (B. 5. p. 4). For the fame reafon m + : N : : m £> : N ; and becaufe there are three magnitudes, c c i 9 4 BOOKV. PROP. XXII. THEOR. and other three, M ^ , m /\ , N , which, taken two and two, have the fame ratio ; .*. ifMjP CZ, =, orZlN then will M + C» => or ^ N , by (B. 5. pr. 20) ; and ,\ V : |:: + : 1 (def. 5). Next, let there be four magnitudes, ■, ^, § , ^ , and other four, £>, ^, > A , which, taken two and two, have the fame ratio, that is to fay, ^p : ♦ - <2> : Q, and : ♦ : : 1 1 : ▲ , then mall ^ : + : : ^ : ▲ ; for, becaufe ■ , ^, , are three magnitudes, and <2>, 0? 5 other three, which, taken two and two, have the fame ratio ; therefore, by the foregoing cafe, <p : j :: ^ : .•■, but I : ♦ :: : ▲ ; therefore again, by the firfl cafe, ^p : ^ : : ^> : ▲ ; and {o on, whatever the number of magnitudes be. .*. If there be any number, Sec. BOOK V. PROP. XXIII. THEOR. T 95 F there be any number of magnitudes, and as many others, which, taken two and two in a crofs order, have the fame ratio ; the firji fliall have to the lajl of the firjl magnitudes the fame ratio which the firji of the others has to the lajl of the fame. N.B. — This is ufually cited by the words "ex cequali in proportione perturbatd ;" or " ex aquo perturbato." Firft, let there be three magnitudes, £, Q , || , and other three, ; O ' £ ' which, taken two and two in a crofs order, have the fame ratio ; o, that is, : O • = and rj : ■ : •♦ then fhall : 1 : : ▲ Let thefe magnitudes and their refpective equimultiples be arranged as follows : — m ,Mrj, w |,M t,«0» w #» then * : Q :: M : M Q (B. 5. pr. 15) ; and for the fame reafon but J :Q ::<> :# (hyp.), jo6 BOOK V. PROP. XXIII. THEOR. .-. M : MQ ::<> : • (B. 5. P r. n); and becaufe O : H :: : 0» (hyp.), ,\ M Q : m ■ :: : w £> (B. 5. pr. 4) ; then, becaufe there are three magnitudes, M ,MO,«|, and other three, M , m £>, m £, which, taken two and two in a crofs order, have the fame ratio ; therefore, if M CZ, =, or 3 m I? then will M C =, or ^ w (B. 5. pr. 2 1 ), and /. : ■ :: : # (B. 5. def. 5). Next, let there be four magnitudes, >p,o, ■• #1 and other four, <^>, %, Hi, Jk., which, when taken two and two in a crofs order, have the fame ratio ; namely, and then fhall V :D : :m : D ■ : :#: ■ ^L •O: * •+' :0: For, becaufe , Q , | | are three magnitudes, BOOKV. PROP. XXIII. THEOR. 197 and , ■§, ▲, other three, which, taken two and two in a crofs order, have the fame ratio, therefore, by the firft cafe, >:!!::#:▲, but ■ : < :: £> : #, therefore again, by the firft cafe, I : < :: <^) : A J and fo on, whatever be the number of fuch magnitudes. .*. If there be any number, &c. iq8 BOOK V. PROP. XXIV. THEOR. j]F the firjl has to the fecond the fame ratio which the third has to the fourth, and the fifth to the fecond the fame which the fix th has to the fourth, the firjl and fifth together jhal I have to the fecond the fame ratio which the third and fix th together have to the fourth. First. Second. Third. Fourth. V D ■ ♦ Fifth. Sixth. o • Let jp : Q : : ■: , and <3 : Q : : • : ►» then ^ + £> : O ::■+#:♦ For <2> : D :: # : ( h yP-)' and [J : ^ : : : ■ (hyp-) and (invert.), .\ <2>: qp :: #: ■ (B. 5. pr. 22); and, becaufe thefe magnitudes are proportionals, they are proportionals when taken jointly, .'. V+6'O'- •+■= #(B. 5 .pr. 18), but o>: D - # : ( h yp-) 5 .'. ¥ + O^O- •+ ■• (B. 5. pr. 22). ,\ If the firft, &c. BOOK V. PROP. XXV. THEOR. 199 F four magnitudes of the fame kind are propor- tionals, the great ejl and leaf of them together are greater than the other two together. Let four magnitudes, ^ -|- ^, I | -|- , ^J, and of the fame kind, be proportionals, that is to fay, * + 0: ■ + :iQ:f and let ■ -|- (3) be the greateft of the four, and confe- quently by pr. A and 14 of Book 5, is the leaft ; then will Jp-f-Q-l- beCB+ + U J becaufe ^ -f- Q : £ -j- : : Q : •*• V ' M-- W+O- ■+ (B. 5 .pr. 19), bu < * + D C ■ + (hyp.), .'• * 1= B(B- 5-pr-A); to each of thefe add ^J -}- , ••• * + D + t= «+D+ ♦• .*. If four magnitudes, &c. 2oo BOOK V. DEFINITION X. DEFINITION X. When three magnitudes are proportionals, the firfl is faid to have to the third the duplicate ratio of that which it has to the fecond. For example, if A, , C , be continued proportionals, that is, A : : : : C , A is faid to have to C the dupli- cate ratio of A : B ; or — rz the fquare of—. This property will be more readily feen of the quantities a ** > > J » for a r"' : . : : : a ', ar 1 r ! and — rr r* — the fquare of — sr r, or of , , j - ; for — - s-jZ= the fquare of— =— . a r ' r DEFINITION XI. When four magnitudes are continual proportionals, the firft is faid to have to the fourth the triplicate ratio of that which it has to the fecond ; and fo on, quadruplicate, &c. increafing the denomination ftill by unity, in any number of proportionals. For example, let A,B, C, D, be four continued propor- tionals, that is, A : i> :: : C :: C : D; A is faid to have to D, the triplicate ratio of A to B ; or - s; the cube of—. BOOK V. DEFINITION XL 20I This definition will be better underftood, and applied to a greater number of magnitudes than four that are con- tinued proportionals, as follows : — Let a r' , , a r , a, be four magnitudes in continued pro- portion, that is, a r 3 : :: '• a r '•'• a r '• < l > then - — =r r 3 ~ the cube of — = r. a Or, let ar 5 , ar*, ar 3 , ar', ar, a, be fix magnitudes in pro- portion, that is ar 5 : ar* : : ar* * ar 3 :: ar 3 : ar 1 : : ar : ar :: ar : a, ar* cir° then the ratio — ■ =: r° z= the fifth power of — ; =: r. a r ar* Or, let a, ar, ar 2 , ar 3 , ar 4 , be five magnitudes in continued proportion ; then — -. =-7 — the fourth power of — — -. r r ar* r* r ar r DEFINITION A. To know a compound ratio : — When there are any number of magnitudes of the fame kind, the firfr. is faid to have to the laft of them the ratio compounded of the ratio which the firft has to the fecond, and of the ratio which the fecond has to the third, and of the ratio which the third has to the fourth ; and fo on, unto the laft magnitude. For example, if A, B, C, D, be four magnitudes of the fame kind, the firft A is faid to have to the laft D the ratio compounded of the ratio of A to B , and of the ratio of B to C, and of the ratio ofC to D ; or, the ratio of DD A B C D E F G H K. L M N 202 BOOK V. DEFINITION A. A to 1 ' is faid to be compounded of the ratios of ' to , H to ( , and I to I > . And if \ has to I. the fame ratio which I has to I , and B to C the fame ratio that G has to H, and ( to h the fame that K has to L ; then by this definition, \ is said to have to L) the ratio compounded of ratios which are the fame with the ratios of E to F, G to H, and K to L« And the fame thing is to be underftood when it is more briefly expreffed by faying, \ has to D the ratio compounded of the ratios of E to F, G to H, and K to | . In like manner, the fame things being iuppofed ; if has to the fame ratio which has to I ' , then for fhort- nefs fake, is faid to have to the ratio compounded of the ratios of E to F, G to H, and K to L. This definition may be better underftood from an arith- metical or algebraical illuftration ; for, in fact, a ratio com- pounded of feveral other ratios, is nothing more than a ratio which has for its antecedent the continued product of all the antecedents of the ratios compounded, and for its confequent the continued product of all the confequents of the ratios compounded. Thus, the ratio compounded of the ratios of : , : ,6:11,2:5, is the ratio of - X X 6 X 2 : X X 11 X 5, or the ratio of 96 : 11 55, or 32 : 385. And of the magnitudes A, B, C, D, E, F, of the fame kind, A : F is the ratio compounded of the ratios of A :B, B : C, C: D, J : E, E : F; for A X B X X X E : B X v X X E X F, XX X X E. or xx xexf = "' or the ratio of A : F - BOOK V. PROP. F. THEOR. 203 ATI OS which are compounded of the fame ratios are the fame to one another. Let A : B :: F : G, 5 i ( '•'• r '. ri» ::D::H:K, and ) : E : : I : L. A B C D E F G H K L Then the ratio which is compounded of the ratios of A : B, B : C, C : D, I : E, or the ratio of A : E, is the fame as the ratio compounded of the ratios of F : G, 5 : H, H : K, K : L, or the ratio of F : L. For-i c and- E XXX F "' H' £ . 1' XXX X X XL' F or the ratio of \ : I is the fame as the ratio of F : L. X x e — and .*. j- — The fame may be demonstrated of any number of ratios fo circumftanced. Next, let A : B : : K : L, 1 : C : : i : K, _- ' I ; ; j • rl» ) : E :: F: G. 2o + BOOK V. PROP. F. THEOR. Then the ratio which is compounded of the ratios of A : , : , : , ]) : E, or the ratio of \ : 1 , is the fame as the ratio compounded of the ratios of :L, i : , : H, F : , or the ratio of F :L. For - = - ' and — ss — ; • A X X X X X XF X X X i I- X X X ' and/. - = -, or the ratio of A : E is the fame as the ratio of F : L. ,*. Ratios which are compounded, 6tc. BOOK V. PROP. G. THEOR. 205 F feveral ratios be the fame to fever al ratios, each to each, the ratio which is compounded of ratios which are the fame to the fir Jl ratios, each to each, Jhall be the fame to the ratio compounded of ratios which are the fame to the other ratios, each to each. ABCDEFGH P Q R S T a bed e f g h V W X ^ If \ : B : : a : b and A : B : : P : Q a : b: : : w C :D .: c : d C:D::Q: R c:d: : W: X E :F :: e :f E : F : : R S e :/: : : : Y and G : II ::g:h G:H :: S T g:h: : : Z then P : T = ' T- A a For .7 = I = 7 z^z 9 _ C c . R D d =: ~~ ' e s ■— F — / = X 3 G y T H h = > ■ x s x x X X X • Q x R x - x r — J p and .*. - ^ - X - > X X z* rP :T = : Z. .*. If feveral ratios, &c. 2o6 BOOK V. PROP. H. THEOR. F a ratio which is compounded of feveral ratios be the fame to a ratio which is con pounded of fever al other ratios ; and if one of the firjl ratios, or the ratio which is compounded of feveral of them, be the fame to one of the laft ratios, or to the ratio which is com- pounded of Jeveral of them ; then the remaining ratio ofthefirji, or, if there be more than one, the ratio compounded of the re- maining ratios, Jliall be the fame to the remaining ratio of the la/i, or, if there be more than one, to the ratio compounded ofthefe remaining ratios. A B C D E F G H P Q R S T X Let A :B, B :C, C :D, D : E, E : F, F : G, G :H, be the nrft ratios, and P : Q^Qj_R, R : S, S : T, T : X, the other ratios ; alfo, let A : H, which is compounded of the rirft ratios, be the fame as the ratio of P : X, which is the ratio compounded of the other ratios ; and, let the ratio of A : E, which is compounded of the ratios of A : B, B :C, C :D, D :E, be the fame as the ratio of P : R, which is compounded of the ratios P : Q^ Qj R. Then the ratio which is compounded of the remaining firft ratios, that is, the ratio compounded of the ratios E : F, F : G, G : H, that is, the ratio of E : H, mail be the fame as the ratio of R : X, which is compounded of the ratios of R : v , S : T, T : X, the remaining other ratios. BOOK V. PROP. H. THEOR. 207 B ecau f e XEXl XDXEXi-XG P X <J X k X S X T B X C X D X E X F X G X H Q X K X - X I X \' or xiiXiXd w e x } x b x g w B XC X D X E * FX(,XH — DXB * X (, X H ' ^ SXTXX' and A X B X C X_D LXS, B X l X D X E — O X R ' . E X F X G R X X ' * F X G XH SXTXX* . E __ /. E : H = R : X. .*. If a ratio which, &c. 2o8 BOOK V. PROP. K. THEOR. F there be any number of ratios, and any number of other ratios, fuch that the ratio which is com- pounded of ratios, which are the fame to the firji ratios, each to each, is the fame to the ratio which is compounded of ratios, which are the fame, each to each, to the laji ratios — and if one of the fir Jl ratios, or the ratio which is compounded of ratios, which are the fame to feveral of the firft ratios, each to each, be the fame to one of the lajl ratios, or to the ratio which is compounded of ratios, which are the fame, each to each, to feveral of the lajl ratios — then the re- maining ratio of the firji ; or, if there be tnore than one, the ratio which is compounded of ratios, which are the fame, each to each, to the remaining ratios of the firji, Jhall be the fame to the remaining ratio of the lajl ; or, if there be more than one, to the ratio which is compounded of ratios, which are the fame, each to each, to thefe remaining ratios. h k m n s AB, CD, EF, GH, KL, MN, a b c d e f g p,i , , y, h k I m n p abed e f g Let A:B, C:D, E:F, G:H, K:L, M:N, be the firft ratios, and : , : , : , ^:W, \ : , the other ratios ; and let A :B — a :b, C :D — b :c, E :F — ■ :,!, G H — d J K L — e :/, M :N . r :?• BOOK V. PROP. K. THEOR. 209 Then, by the definition of a compound ratio, the ratio of a : ~ is compounded of the ratios of a 'b> b '-c> c : d> d -e* g'.f, f'gt which are the fame as the ratio of A '• B> C : D» E*: F, G : H, K : L, M : N, each to each. :P — h :k, QJR — k :/, : — I: m, :W — m : n, :Y — n •p. Then will the ratio of h :p be the ratio compounded of the ratios of // : k, k\l, l\m, m\n, n:p, which are the fame as the ratios of : , : , } :T»-V :W> X *Y » each to each. ,*. by the hypothefis a : » = h :p. Alfo, let the ratio which is compounded of the ratios of A : B , C : D , two of the firft ratios (or the ratios of a : Ci for A : B = a : j» and C :D = (, : : ), be the fame as the ratio of a : d, which is compounded of the ratios of a : b, b : c, c : d, which are the fame as the ratios of 1 : p , > • R » J! '• 1 » three of the other ratios. And let the ratios of h : s, which is compounded of the ratios of h : k, k : m, m : n, n : s, which are the fame as the remaining firft ratios, namely, E :F» G :H> K =L» M : N ; alfo, let the ratio of e : g, be that which is com- pounded of the ratios e : f, f : g, which are the fame, each to each, to the remaining other ratios, namely, r :W, [ : Y « Then the ratio of h : s fhall be the fame as the ratio of e : g ; or h : s — e : g. p A XC XE X»: XK XM „ X b X c X J X e X / B X D X F X H XI, X N T b X c X J X •• X / X g ' E E 2io BOOK V. PROP. K. THEOR. . X X XX h X k X I XmXn X X X X JX'X«X»X?' by the compofition of the ratios ; aX CXcXdX e X f h X k X I X m X n ,, x bXcXdXtX/Xg kX I XmX * Xp ^y p -)> or g Xi V c X c/ X e X / h X k X I w m X » iXc ^ dX e X/Xg k X I Xm *> n Xp' but ° X * r= A X " = X X __ a X b X c __ h Xk X I . t X i BXD "X X bXcXti " * X / X «» ' . cXdXt X ) m X n ' ' dXt X fXs " Xp' A „ j C X c X t X f li X k X m X n ,, , d Xt x/xg k X m X n X s and TO X » " X /> e f X f Xg (hyp.), • h X k X m X n . — e f • I k X m X r X s " <V • • • h s — e .'. h : s — e : g- '. If there be any number, &c. Algebraical and Arithmetical expositions of the Fifth Book of Euclid are given in Byrne's Doctrine of Proportion ; published by Williams and Co. London. 1841. BOOK VI. DEFINITIONS. I. ECTILINEAR figures are faid to be fimilar, when they have their fe- veral angles equal, each to each, and the fides about the equal angles proportional. II. Two fides of one figure are faid to be reciprocally propor- tional to two fides of another figure when one of the fides of the firft is to the fecond, as the remaining fide of the fecond is to the remaining fide of the firft. III. A straight line is faid to be cut in extreme and mean ratio, when the whole is to the greater fegment, as the greater fegment is to the lefs. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to its bafe, or the bafe produced. 2 ;2 BOOK VI. PROP. I. THEOR. PUR I ANGLES and parallelo- grams having the fame altitude are to one another as their bafes. I and A Let the triangles have a common vertex, and their bafes — — and ■ -»-—» in the fame ftraight line. Produce — — — — — both ways, take fuccemvely on 1 ' produced lines equal to it ; and on — — • pro- duced lines succeffively equal to it ; and draw lines from the common vertex to their extremities. A The triangles 4lJZ. wi thus formed are all equal to one another, fince their bafes are equal. (B. i. pr. 38.) A and its bafe are refpectively equi- i multiples of m and the bafe BOOK VI. PROP. I. THEOR. 2< 3 Lk In like manner m _ and its bale are refpec- { tively equimultiples of ^ and the bafe ■ ■* . .*. Ifwor6times jf |~ = or ^ n or 5 times ■> then ;« or 6 times ■ CZ == or Z] 8 or 5 times wu« , w and « fland for every multiple taken as in the fifth definition of the Fifth Book. Although we have only mown that this property exifts when m equal 6, and n equal 5, yet it is evident that the property holds good for every multiple value that may be given to m, and to n. a (B. 5. def. 5.) Parallelograms having the fame altitude are the doubles of the triangles, on their bafes, and are proportional to them (Part 1), and hence their doubles, the parallelograms, are as their bafes. (B. 5. pr. 15.) Q. E. D. 2I 4 BOOK VI PROP. II. THEOR. * * A F a Jlraight line be drawn parallel to any jide .—.—..— of a tri- angle, it Jliall cut the other fides, or thofe Jides produced, into pro- portional fegments . And if any Jlraight line divide the fides of a triangle, or thofe fides produced, into proportional feg- ments, it is parallel to the remaining .fide— -• Let PART I. ., then fhall • ■••«■»•■» D raw V- and . (B. i.pr. 37); \ (B.5.pr. 7 );but *■*»»■■ ■ (B. 6. pr. i), ■**«•!•■ HiHtllllB' . (B. 5 .pr. ii). BOOK VI. PROP. II. THEOR. 2] 5 PART II. Let ■■■■•■■■a then Let the fame conftrudtion remain, becaufe and -------- ? «*«■■■!■■■ * > (B. 6. pr. i); but !■*■»*■ t I I » I (hyp-), ■7 ■ ♦. • • \ (B. 5- P r - 1 1 ,7= 3 ( B -5-P>'-9); but they are on the fame bafe -■■■■••■■• and at the fame fide of it, and - II (B- i-pr. 39). Q. E. D. 2l6 BOOK VL PROP. III. THEOR. RIGHT line ( ) bifecling the angle of a triangle, divides the op- pojite Jide into fegments — ----- ) proportional to the conterminous Jides (. )• And if a Jlraight line (• ) drawn from any angle of a triangle divide the oppofte fde (—^— ■■■■■■) into fegments ( 9 ...-......) proportional to the conterminous fdes (—■■■», ___ ), it bifecls the angle. PART I. Draw -■■••-■•» | — — — , to meet „„ ; then, := ^ (B. i. pr. 29), (B. 1. pr. 6); and becaufe (B. 6. pr. 2); (B. 5. pr. 7 ). BOOK VI. PROP. III. THEOR. 217 PART II. Let the fame conftrudlion remain, and — — — : .— — — . :: 1 : .--—-- (B. 6. pr. 2); but — — — : — — — :: 1 : ■ (hyp-) (B. 5. pr. n). and .*. •■■■«■■■- — (B. 5. pr. 9), and .*. ^f — ^ (B. 1. pr. 5); but fince II ; m = t and zr ^f (B. 1. pr. 29); .". ^ = T, and = J^. and .'. .ii ■■ biiedts J^ , Q.E. D. F F 2l8 BOOK VI. PROP. IF. THEOR. N equiangular tri- angles ( S \ and \ ) the fides rt^o«/ /7j^ ^«^/ angles are pro- portional, and the Jides which are ^L oppojite to the equal angles are - .1 ■ S ■ - - - homologous Let the equiangular triangles be fo placed that two fides oppofite to equal angles and ^^^ may be conterminous, and in the fame ftraight line; and that the triangles lying at the fame fide of that ftraight line, may have the equal angles not conterminous, i. e. jtKk oppofite to , and to Draw and . Then, becaufe * * --^— ■— | , . «...— (B.i.pr.28); and for a like reafon, •■■«■»■■■« 1 1 — — , is a parallelogram. But (B. 6. pr. 2); BOOK VI. PROP. IV. THEOR. 219 and fince = -^—^— (B. 1. pr. 34), — : :: : — ; and by alternation, (B. 5. pr. 16). In like manner it may be mown, that a> •• •«■■*••» and by alternation, that * * a • • • a • a 1 a • a> but it has been already proved that am lamiiiaii and therefore, ex xquali, (B. 5. pr. 22), therefore the fides about the equal angles are proportional, and thofe which are oppofite to the equal angles are homologous. Q. E. D. 220 BOOK VI. PROP. V. THEOR. F two triangles have their Jides propor- tional (••■■«■■- : ........-_ :: — m—m l — ) and ( «■■!■■■■■■■■ •. » ■ : : — — : — — «— ) //z^ art- equiangular, and the equal angles are Jubt ended by the homolo- gous Jides. From the extremities of 9 and , making W= M (B. ,. pr. 23); and confequently ^ = (B. I. pr. 32), and fince the triangles are equiangular, draw (B. 6. pr. 4); but (hyp-); and confequently (B. 5. pr. 9 ). In the like manner it may be mown that BOOK VI. PROP. V. THEOR. 221 Therefore, the two triangles having a common bafe , and their fides equal, have alfo equal angles op- ^ = , ¥a„d/l = W pofite to equal fides, i. e. s\ m (B. 1. pr. 8). But ^F = ^fc (conft.) and .*. z= mtk ; for the lame reafon m \ := m A ■ and confequently £^ z= (B. 1. 32); and therefore the triangles are equiangular, and it is evi- dent that the homologous fides fubtend the equal angles. Q. E. D. 2 22 BOOK VI. PROP. VI. THEOR. s\ F two triangles ( ^S Z\ " and ) have one \ angle ( ^Kk ) of the one, equal to one A \ angle ( f \ ) of the other, and the fides ^L about the equal angles proportional, the ,,,,1^ triangles /hall be equiangular, and have thofe angles equal which the homologous fides fubt end. From the extremities of of ^^— • , one of the fides S \ , about m \ 9 draw — — . and , making ▼ = A , and ^F — J^ ; then ^ = (B. i. pr. 32), and two triangles being equiangular, >•>■>■>■■■■> -- (B. 6. pr. 4); but (hyp.) ; «**••••••»•<• (B. 5. pr. 11), and confequently = — (B. 5. pr. 9); BOOK VI. PROP. VI. THEOR. 223 z\ = \/ in every refpedt. (B. 1. pr. 4). But ^J == j^ (conft.), and .*. / ■ ZZ J^ : and fince alio ■ \ — mtk . ' \ ■=. (B. 1. pr. 32); y\ and .*. jf„.„^k and -^ \ are equiangular, with their equal angles oppoiite to homologous fides. Q^E. D. 224 BOOK VI. PROP. VII. THEOR. / .♦ 'V F two triangles ( A and * ) have one angle in each equal ( equal to ^| ), the Jides about two other angles proportional 4 and each of the remaining angles ( and ^-J| ) either lefs or not lefs than a right angle y the triangles are equiangular, and thofe angles are equal about which the fides are proportional. Firft let it be affumed that the angles ^^ | and <^ are each lefs than a right angle : then if it be fuppofed that itA and ^A contained by the proportional fides, are not equal, let ^^,\ be the greater, and make Becaufe 4 = * ( h yP-)> and ^A = ^\ (conft.) = ^^ (B. I. pr. 32); BOOK VI. PROP. VII THEOR. 225 M ■ ■■«■■■«■■• (B. 6. pr. 4), but — ^— : :: : (hyp-) (B. 5. pr. 9), and .*. '^^ = ^^ (B. 1. pr. 5). But ^^ I is lefs than a right angle (hyp.) .*. 4^ is lefs than a right angle ; and .*. mull be greater than a right angle (B. 1 . pr. 13), but it has been proved = ^^ and therefore lefs than a right angle, which is abfurd. .*. ^^\ and ^->\ are not unequal ; .*. they are equal, and fince "^B — \ (hyp.) 4 = 4 (B. 1. pr. 32), and therefore the tri- angles are equiangular. ^ and ^ But if ^^B and ^--^ be affumed to be each not lefs than a right angle, it may be proved as before, that the triangles are equiangular, and have the fides about the equal angles proportional. (B. 6. pr. 4). Q. E. D. G G :26 BOOK VI. PROP. VIII. THEOR. N a right angled triangle )>if — ) ( a perpendicular ( be drawn from the right angle to the oppojitejide, the triangles f ^/j^ | j |^ ) on each Jide of it are fimilar to the whole triangle and to each other. Becaufe common to (B. i. ax. 1 1), and and A = 4 (B. i. pr. 32); and are equiangular ; and coniequently have their fides about the equal angles pro- portional (B. 6. pr. 4), and are therefore fimilar (B. 6. def. 1). In like manner it may be proved that nk is fimilar to L ; but has been lliewn to be fimilar to k and |L are 9 • • fimilar to the whole and to each other. Q. E. D. BOOK VI. PROP. IX. PROB. 22" ROM a given jlraight line ( ) to cut off any required part . From either extremity of the given line draw ——»"••■... making any angle with * and produce ■•■•■■I till the whole produced line ■••■••■■•■ contains " as often as contains the required part. Draw >, and draw II is the required part of For fince (B. 6. pr. 2), and by compofition (B. 5. pr. 18) ; but ■•-•-"» contains as often as contains the required part (conft.) ; is the required part. Q. E. D. 228 BOOK VI. PROP. X. PROB. and draw (' O divide a Jlraight line ( — ) fimilarly to a given divided line )• From either extremity of the given line — draw -■—••..—-■-.«. making any angle ; take ............ .......... an( j ■■•■a««a«4 equal to refpedively (B. i. pr. 2) ; , and draw —■--■■■» and — — II to it. or and Since j —■■■-««- \ are (B. 6. pr. 2), (B. 6. pr. 2), and .*. the given line fimilarly to (conft.), (conft.), is divided Q.E. D. BOOK VI. PROP. XI. PROB. 229 O find a third proportional to two given Jiraight lines At either extremity of the given line -^^— ^ draw ..-— making an angle ; take ......... r= , and draw ■ ; make ........ ~ and draw ........ (B. 1. pr. 31.) .■ uj-jj is the third proportional to — — — and . For fince ...■a n..i..« (B. 6pr. 2); but — ■(conft.) ; (B. 5. pr. 7). Q^E. D. 2 3 BOOK VI. PROP. XII. PROB. O find a fourth pro- portional to three given lines and take and alfo draw and I •«««»«•* !«»»«•«««« Draw making any angle ; (B. i. pr. 31); is the fourth proportional. 1 Y mt J On account of the parallels, (B. 6. pr. 2); ■— } = {: Er-}( conft -) ; *. ■■■■■■■■■■■ * »■•■••»■«■ jj •» (B. 5. pr. 7). Q^E. D. BOOK VI. PROP. XIII. PROB. 2 3* O find a mean propor- tional between two given Jlraight lines }• Draw any ftraight line make — ■ and ; bifecl and from the point of bifedtion as a centre, and half the line as a radius, defcribe a femicircle draw — ■— «— — — — . r\ is the mean proportional required. Draw and Since ^| p> is a right angle (B. 3. pr. 31), ar >d — ^— is J_ from it upon the oppofite fide, •*• ^~~^ is a mean proportional between — ^— and '■ (B. 6. pr. 8), and .*. between - and (conft.). QE.D 232 BOOK VI. PROP. XIV. THEOR. QJJ A L parallelograms \ and which have one angle in each equal, have the Jides about the equal angles reciprocally proportional II. And parallelograms which have one angle in each equal, and the Jides about them reciprocally proportional, are equal. Let and and J and and — ■ — , be (o placed that . — — may be continued right lines. It is evi- dent that they may affume this pofition. (B. i. prs. 13, 14, '5-) Complete % Since V V:\:\ [B. 5. pr. 7.) BOOK VI PROP. XIV. THEOR. 233 (B. 6. pr. 1.) The fame conftrudtion remaining (B. 6. pr. 1.) — (hyp.) (B. 6. pr. 1.) (B. 5. pr. n.) and .*. = ^^ (B. 5. pr. 9). Q^E. D. H H 234 BOOK VI. PROP. XV. THEOR. I. QUAL triangles, which have one angle in each equal 1 ^^ = ^m ), have the fides about the equal angles reciprocally proportional II. And two triangles which have an angle of the one equal to an angle of the other, and the Jides about the equal angles reci- procally proportional, are equal. I. Let the triangles be fo placed that the equal angles ^^ and ^B may be vertically oppofite, that is to fay, fo that — — — and — — — may be in the lame ftraight line. Whence alfo — — — — ■ and muft be in the fame ftraight line. (B. i. pr. 14.) Draw ■— — — . then > 4 (B. 6. pr. 1.) (B. 5. pr. 7.) (B. 6. pr. 1.) BOOK VI. PROP. XV. THEOR. 235 A (B. 5. pr. 11.) II. Let the fame conftruction remain, and ^^^r * ' (B. 6. pr. 1.) and A (B. 6. pr. 1.) But : ;: - -: , (hyp.) (B.5 pr. 11); (B. 5. pr. 9.) Q. E. D. • • • A^1 A=A 3 6 BOOK VI. PROP. XVI. THEOR. PART I. F four Jh ■ aight lines be proportional the reclangle ( ■ ) contained by the extremes, is equal to the rectangle X ----- — - ) contained by the means. PART II. And if the re£l- angle contained by the extremes be equal to the reSlangle con- tained by the means, the four Jlraight lines are proportional. PART I. From the extremities of — and -■— ™ «—o^bb» and i _]_ to them and ~ draw and refpe<£tively : complete the parallelograms and I And fince, (hyp-) (conft.) (B. 6. pr. 14), BOOK VI. PROP. XVI. THEOR. 2 37 that is, the redtangle contained by the extremes, equal to the redlangle contained by the means. PART II. Let the fame conltrudtion remain ; becaufe ■ ■a a ■»■***■ vmmwwww and mmm—mm — ....a...... • • ■■BH (B. 6. pr. 14). But = . and — = -— — . (conft.) ■■■■■■■■»■■■■■■ **«■■■■■■•■■• ■■»■■■•■■■■ (B. 5. pr. 7). Q. E. D. -»-■ 2 3 8 BOOK VI. PROP. XVII. THEOR. PART I jF three jlraight lines be pro- portional (■■■ : __ reSlangle under the extremes is equal to the fquare of the mean. PART II. And if the rettangle under the ex- tremes be equal to the fquare of the mean, the three Jlraight lines are proportional. lince then PART I. A flu me X ., and (B. 6. pr. 16). or But X '9 X ___ 2 ; therefore, if the three ftraight lines are proportional, the redtangle contained by the extremes is equal to the fquare of the mean. PART II. Aflume - X ■ . , then X — (B. 6. pr. 16), and Q. E. D. BOOK VI. PROP. XVIII. THEOR. 239 N a given Jlraight line (■ to conjlruB a recJilinear figure Jimilar to a given one and Jimilarly placed. ) Refolve the given figure into triangles by drawing the lines . -. and At the extremities of ^ = f^V and again at the extremities of and ^k — ^ — ^\ T in like manner make * = \/andV = V Th en is fimilar to It is evident from the construction and (B. 1. pr. 32) that the figures are equiangular ; and fince the triangles W and V are equiangular: then by (B. 6. pr.4), :: — and 240 BOOK VI. PROP. XVIII. THEOR. Again, becaule ^ and are equiangular, ^^ md ^B ._ •• * ex aequali, (B. 6. pr. 22.) In like manner it may be fhown that the remaining fides of the two figures are proportional. .-. by (B. 6. def. i.) is fimilar to and fimilarly fituated ; and on the given line Q^E. D. BOOK VI. PROP. XIX. THEOR. 241 IMILAR trian- gles { \ and ^fl B ) are to one another in the duplicate ratio of their homologous Jides. Mk and m Let 4Bt and ^^ be equal angles, and ....--—— and ■ .- homologous fides of the fimilar triangles and ^j m and , and on — ■■■■ the greater of thefe lines take ....... a third proportional, fo that draw (B. 6. pr. 4) ; but (B. 5. pr. 16, alt.), (conft.), — confe- 1 1 242 BOOK VI. PROP. XIX. THEOR. A\ quently = ^* for they have the fides about the equal angles ^^ and 4Bt reciprocally proportional (B. 6. pr. 15); ■AAA\ (B. 5P r. 7); but A \ : ^fc :: — — : (B. 6. P r. 1), Aa • • ..>■>. that is to fay, the triangles are to one another in the dupli- cate ratio of their homologous fides and (B. 5. def. ii). Q^ E. D. BOOK VI. PROP. XX. THEOR. 243 [IMILAR poly- gons may be di- vided into the fame number of fimilar triangles, eachfimilar pair of which are propor- tional to the polygons ; and the polygons are to each other in the duplicate ratio of their homologous /ides . Draw and and and — — — - j refolving the polygons into triangles. Then becaufe the polygons are fimilar, and — — «•■«••■«««• are fimilar, and ^ — ^ (B. 6. pr. 6); but ♦ -♦ = w becaufe they are angles of fimilar poly- gons ; therefore the remainders g/^ and ^k are equal ; hence «■»«•■■■■■ ; ...«...** ;; ._..___._ ; ••■•■•*•••■ on account of the fimilar triangles, 244 BOOK VI. PROP. XX. THEOR. and -.-..-.-- : _^__. •• • _ «. on account of the limilar polygons, ■■■■■■a ■■■■ ex sequali (B. 5. pr. 22), and as thefe proportional fides contain equal angles, the triangles s M ^ and ^^. are limilar (B. 6. pr. 6). In like manner it may be fhown that the triangles ^^ and ^ W are limilar. But is to in the duplicate ratio of •■■■■■■■■. to — — — — (B. 6. pr. 19), and M^- is to ^^ in like manner, in the duplicate ratio of ««■■-■■■■«■ to «— — — ; >> (B. 5. pr. 11); Again M ^^ is to ^^r in the duplicate ratio of M^r to ^^~ ^r ^W to — — — , and is to in BOOK VI. PROP. XX. THEOR. 245 the duplicate ratio of — — - to ■ ▼ ▼ and as one of the antecedents is to one of the confequents, fo is the fum of all the antecedents to the fum of all the confequents ; that is to fay, the fimilar triangles have to one another the fame ratio as the polygons (B. 5. pr. 12). But is to in the duplicate ratio of to ^ is to ™| WL. in the duplicate ratio of _________ to _____ . Q E. D 246 HOOK VI. PROP. XXI. THEOR. ECTILINEAR figures ( and which are fimilar to the fame figure ( are fimilar alfo to each other. Since Hi B^. and arc fimi- lar, they are equiangular, and have the fides about the equal angles proportional (B. 6. def. 1); and fince the figures and ^^ are alfo fimilar, thev are equiangular, and have the fides about the equal angles proportional ; therefore ■■■^ and ■■fet. are alio equiangular, and have the fides about the equal angles pro- portional (B. 5. pr. 1 1), and are therefore fimilar. Q.E. D. BOOK VI. PROP. XXII. THEOR. 247 PART I. Y four fir aight lines be pro- portional (■■■■■■■ : "-™"" :: — : ), the fimilar rectilinear figures Jimilarly described on them are alfo pro- portional. PART 11. And if four Jimilar rectilinear figures, Jimilarly defcribed on four ftraight lines, be proportional, the firaight lines are alfo proportional. Take and — to fince but and part 1. a third proportional to . and •••■•■■■•« a third proportional — and (B. 6. pr. 11); :: :- (hyp.), ... ...... ;; - ; >■■■■■■■■■• (conft.) .*. ex asquali, (B. 6. pr. 20), , ••••••••••• * 248 BOOK VI. PROP. XXII. THEOR. and .*. (B. 5. pr. 11). PART II. Let the fame conftrudlion remain (hyp-). (conft.) (B. 5. pr. 11). Q.E. D. BOOK VI PROP. XXIII. THEOR. 249 QUIANGULAR parallel- ograms ( and ) are to one another in a ratio compounded of the ratios of their jides. Let two of the fides — — _— m and about the equal angles be placed fo that they may form one ftraight line. Since ^ + J = f\\ and J^ = (hyp.), and /. 4 + ■ 11 and <■ — form one ftraight line (B. 1. P r. 14); complete £' . Since # and # (B. 6. pr. 1), ;B.6. pr. 1), has to _. to a ratio compounded of the ratios of , and of «^— ■— a to 1 . K K. Q^E. D. 2 5 o BOOK VI. PROP. XXIV. THEOR. {ELJ) N any parallelogram (^ the parallelograms ( i ^j B: and f I ) which are about the diagonal are Jtmilar to the whole, and to each other. As and B have common angle they are equiangular; but becaufe ■ I and are fimilar (B. 6. pr. 4), ■»•«»■■-■ and the remaining oppofite fides are equal to thofe, .*. B—J and B-L—J have the fides about the equal angles proportional, and are therefore fimilar. In the fame manner it can be demonftrated that the rH and B parallelograms £]_J and f / are fimilar. Since, therefore, each of the parallelograms and B is fimilar to to each BOOK VI. PROP. XXV. PROB. 25 1 O defcribe a rectilinear figure, which /hall be fimilar to a given | rectilinear figure ( equal to another \wb )• ),and Upon defcribe and upon . defcribe | | = tB^ ■ and having zz (B. i. pr. 45), and then and ....... will lie in the fame ftraight line (B. 1. prs. 29, 14), Between and ■• find a mean proportional (B. 6. pr. 13), and upon defcribe , fimilar to and fimilarly fituated. Then For fince and are fimilar, and j .......... (conft.), (B. 6. pr. 20) ; 252 BOOK VI. PROP. XXV. PROB. but : _ M (B.6.pr. i); ;B. 5 .pr.n); but and .\ and (conft.), (B. 5. pr. 14); = ■} (conft.) ; consequently, which is fimilar to ^^fl Bk is alio = w . Q. E. D. BOOK VI. PROP. XXVI. THEOR. 253 F fimilar and Jimilarly pofited parallelograms U and (/J) have a common angle, they are about the fame diagonal. ( For, if poffible, let be the diagonal of draw (B. 1. pr. 31 Since diagonal L.*13 are about the fame A and have common, they are fimilar (B. 6. pr. 24) ; but (hyp-)» and .*. (B. 5. pr. 9.), which is abfurd. is not the diagonal of a in the fame manner it can be demonftrated that no other line is except ===== . Q. E. D. 2 54 BOOK VI. PROP. XXVII. THEOR. F all the rectangles contained by the fegments of a given Jlraight line, the greatejl is the fquare which is defcribed on half the line. be the unequal fegments, equal fegments ; For it has been demonftrated already (B. 2. pr. 5), that the fquare of half the line is equal to the rectangle con- tained by any unequal fegments together with the fquare of the part intermediate between the middle point and the point of unequal fection. The fquare defcribed on half the line exceeds therefore the rectangle contained by any un- equal fegments of the line. Q.E. D. BOOK VI. PROP. XXVIII. PROB. 255 O divide a given Jlraight line ( ) J fo that the rec- tangle contained by its segments may be equal to a given area, m not exceeding the fauare of half the line. Let the given area be = Bifedt «.»— make ••"* and if -— »» or the problem is iblved. But if muft % then (hyp.)- Draw make - with ^" •«■»•■»■» or as radius defcribe a circle cutting the given line ; draw Then X (B. 2. pr. 5.; - But + (B. 1. pr. 47); 256 BOOK VI. PROP. XXVIII. PROB. .-. X + ' « I * j from both, take — — — and -—- X that X __ 4 But i— — mm» — (conrt.), and .*. «■■ i —■«■»» is fo divided — « Q^E. D. BOOK VI. PROP. XXIX. PROB. 257 O produce a givenjlraight line (— ),fo that the recJangle con- tained by the fegments between the extremities of the given line and the point to which it is pro- duced, may be equal to a given area, i. e. equal to the fquare on Make — , and draw — -■«— draw with the radius meeting Then — — * and ■, defcribe a circle ■ produced. X But 1 (B. 2. pr. 6.) = \ ......... 9 _|_ _ 2 (B. 1. pr.47.) and * + from both take X but = + 2 « = the given area. C^E. D. L L 258 BOOK VI. PROP. XXX. PROB. O cut a given finite jiraight line (— — ) in extreme and mean ratio. On (B. i. pr. 46) ; and produce — x defcribe the fquare — — — , (o that (B. 6. pr. 29); take and draw «■ meeting II Then u X- (B. 1. pr. 31). »■■■■■■» and is /. — 1 • and if from both thefe equals be taken the common part J , which is the fquare of " will be zr II, which is ^ ■■■- X that is X and ■•■■••■ is divided in extreme and mean ratio. (B. 6. def. 3). Q. E. D. BOOK VI. PROP. XXXI. THEOR. 259 F any fimilar rectilinear figures be fimilar ly defer ibed on the fides of a right an- gled triangle ( /\ ), the figure defer ibed on the fide ( ) fub- tending the right angle is equal to the fum of the figures on the other fides. From the right angle draw- to — then ••■»■ : .- (B. 6. pr. 8). (B. 6. pr. 20). perpendicular but Hence but (B. 6. pr. 20). »+ + + ■ as»Mt»»m ifiiiuai and ,\ Q. E. D. 260 BOOK VI. PROP. XXXII. THEOR F two triangles ( A ^ tf «^ /%\ ), have two Jides pro- ^k W^mtSn portionai ( ; — — ~ \\ :: .......... ; «.... ), and be Jo placed \ at an angle that the homologous Jides are pa- rallel, the remaining Jides ( one right line. and ) Jl orm Since = (B. i. pr. 29) ; and alfo fince — — — | •• •• = A (B. 1. pr. 29); = ^^ ; and fince (hyp-). the triangles are equiangular (B. 6. pr. 6) ; A = A but — ▲+ +A=±+4+A m (B. 1. pr. 32), and .*. and lie in the fame flraight line (B. 1. pr. 14). Q.E. D. BOOK VI. PROP. XXXIII. THEOR. 261 N equal circles ( 00 ), angles, whether at the centre or circumference, are in the fame ratio to one another as the arcs on which they Jland ( fo alfo are feci or s. L:4-.- ) i o Take in the circumference of | I any number of arcs «■—» , — , &c. each 35 ■— ? and alfo in the circumference of I take any number of O arcs , &c. each rr , draw the radii to the extremities of the equal arcs. Then fince the arcs " 9 "■"-• . ■—■-, &c are all equal, the angles # , W , \, &c. are alfo equal (B. 3. pr. 27); .*. mwk is the fame multiple of which the arc is of «« ; and in the fame manner 4B^ 4. is the fame multiple of ml , which the arc ,»„„„„«»•*** is of the arc 262 BOOK VI. PROP. XXXIII. THEOR. Then it is evident (B. 3. pr. 27), if 4V ( or if w times ^ ) EZ> =, ^| 4W- (or « times ^ ) then ^-^ ^+ (or m times «*— ) C> =^> ~ *•••• (o r « times ).; • • ^» ^^ • • ^^^ • -1... - ^ (B. 5. def. 5), or the angles at the centre are as the arcs on which they ftand ; but the angles at the circumference being halves of the angles at the centre (B. 3. pr. 20) are in the fame ratio (B. 5. pr. 15), and therefore are as the arcs on which they ftand. It is evident, that fedtors in equal circles, and on equal arcs are equal (B. 1. pr. 4; B. 3. prs. 24, 27, and def. 9). Hence, if the fectors be fubftituted for the angles in the above demon ftration, the fecond part of the propofition will be eftablifhed, that is, in equal circles the fedlors have the fame ratio to one another as the arcs on which they fland. Q.E. D. angle Z BOOK VI. PROP. A. THEOR. F the right line (........), bifeSling an external angle ll ij/' the trz- meet the oppojite 263 fide ( ) produced, that whole produced fide ( ), and its external fegment (-—-«•-) will be proportional to the fides ( —...- #»</ ), w^/'c/i contain the angle adjacent to the external bifecJed angle. For if ■■ be drawn || >*-— •->«« , \ /, (B. 1. pr. 29) ; = ^,(hy P .). = , (B. 1. pr. 29); and , (B. i.pr. 6), and (B. 5. pr. 7); But alfo, IBIIIIEIIII- *iaani>f • ■ ■■ (B. 6. pr. 2); and therefore (B. 5 .pr. 11). Q. E. D. 264 BOOK VI. PROP. B. THEOR. F an angle of a triangle be bi- fecJed by a Jlraight line, which likewife cuts the bafe ; the rec- tangle contained by the fides of the triangle is equal to the rectangle con- tained by the fegments of the bafe, together with the fquare of the Jlraight line which b/fecJs the angle. Let be drawn, making ^ = £t ; then fhall ... x + o (B. 4. pr. 5), produce to meet the circle, and draw — Since 4fl = 4Hk (hyp-)> and = ^T (B. 3. pr. 21), ,ZL \ are equiangular (B. 1. pr. 32) ; y (B. 6. pr. 4); ROOK VI. PROP. R. THEOR. 265 • • — — — X ~~^^~ — x (B.6. pr. 16.) = -X + f (B. 2. pr. 3); but X = ------. w (B. 3. pr. 35 ); X = X h Q.E. D. M M 266 BOOK VI. PROP. C. THEOR. mall F from any angle of a triangle a Jlraight line be drawn perpendi- cular to the bafe ; the reft angle contained by the fides of the tri- angle is equal to the rectangle contained by the perpendicular and the diameter of the circle defcribed about the triangle. From draw _L ■ X = of >** „*7-> _ • s •«•■««* — ; then Xthe diameter of the defcribed circle. Defcribe O (B. 4. pr. 5), draw its diameter and and draw — — ; then becaufe — ■ >> (confl. and B. 3. pr. 31); = /^ (B. 3. pr. 21); • j*. is equiangular to A (B. 6. pr. 4); and .*. --• X == X (B. 6. pr. 16). Q.E. D. BOOK VI. PROP. D. THEOR. 267 HE rectangle contained by the diagonals of a quadrilateral figure infcribed in a circle, is equal to both the rectangles contained by its oppojite Jides. Let / figure infcribed in be any quadrilateral o and draw and then X Make 4^ = W (B.i.pr. 23), ^ = ^ ; and = (B. 3. pr. 21); and .*. (B. 6. pr. 4); X amiiiiB (B. 6. pr. 16) ; again, becaufe 4» = W (conft.), X 268 BOOK VI. PROP. D. THEOR. and \/ = \y (B. 3. pr. 21) (B. 6. pr. 4); and /. <•--...•••* ^ «.»— — ; mi ■■■•■!■■■» (B. 6. pr. 16); but, from above, X = X + X (B. 2. pr. 1 . Q. E. D. THE END. cmiswick: PRiNTrn by c. whitti.noiiam. /- sU jS ^r, j^zser. ^5^- '^Z&r ^^ - &&^> 4>Ts jjpF' ^gS ^5g . ^ ■ j^^ J3H ^ ■ y'yj t if mjfA Ulti yr - r 4 I SJ7